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Question 261693: Use the five-step problem solving strategy to find the measure of the angle described.
The angle's measure is 60degrees more than that of its complement.
I am having problems with setting it up please help

Answer by ikleyn(53748)

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You can put this solution on YOUR website!
.
Use the five-step problem solving strategy to find the measure of the angle described.
The angle's measure is 60degrees more than that of its complement.
I am having problems with setting it up please help
~~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution in the post by @mananth is incorrect due to his arithmetic errors.
I came to bring a correct solution.

Let the angle be x
the complement will be 90-x
x = (90-x) + 60
x = 90 + 60 - x
2x = 150
x = 150/2 = 75 degrees
the complement will be 15 degrees.

Solved correctly.

Question 1179536: Hello! Can you help me solve this?

“If angle a and angle b are vertical angles and angle a is 90 degrees, what is angle B?
Answer by n2(79)

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You can put this solution on YOUR website!
.

Vertical angles are congruent and have equal measures.


So, if angle A is 90 degrees, then vertical angle B is 90 degrees, too.

Solved and explained.

The plot and the writing in the post by @mananth are irrelevant to the problem,
so you can ignore them.

Question 1179537: Could anyone help me with the next questions I sent.
If angle a and angle b is same-side interior angles, and a is 30 degrees, what is angle B?
Found 3 solutions by n2, math_tutor2020, ikleyn:
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.

The same side interior angles, formed by two parallel line and a transversal line, always sum up to 180 degrees. So we write A + B = 180, 30 + B = 180, B = 180 - 30 = 150 degrees. ANSWER. B = 30 degrees.

Solved and explained.

The plot in the post by @mananth is irrelevant to this problem,
so you better ignore it for the safety of your mind.

Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

Same-side interior angles are on the same side of the transversal line, and also inside the parallel likes like so

Since we have parallel lines, the same side interior angles are supplementary. They add to 180 degrees.
a+b = 180
b = 180-a
b = 180-30
b = 150

Tutor mananth would be correct if a,b were alternate interior angles

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Could anyone help me with the next questions I sent.
If angle a and angle b is same-side interior angles, and a is 30 degrees, what is angle B?
~~~~~~~~~~~~~~~~~~~~~~

        The plot in the post by @mananth is irrelevant to the problem.
        In other words, it is wrong plot.
        So, his answer a = b is INCORRECT and must be denied.

If 'a' and 'b' are same side interior angles, it means that a + b = 180 degrees. It means that in this problem 30 + b = 180 degrees. Hence, b = 180 - 30 = 150 degrees. ANSWER

At this point, the problem is solved correctly and completely.

Question 1179538: Here’s the last question I need help on. Thank you for anyone who solved this!!

If angle a and angle b are alternate interior angles solve for X given:
A = 2x and B = x+ 30

Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.

The alternate interior angles, formed by two parallel lines and a transverse line, are congruent and have the same measure. So, we write 2x = x + 30 and obtain immediately from this equation 2x - x = 30, x = 30 degrees. <<<---=== ANSWER

Solved.

Question 1201148: How do you find what X and y are if they are in the same angles together
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
How do you find what X and y are if they are in the same angles together
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As I read this post, it seems to me that the visitor wants to scare a reader.

Question 716555: the supplement of an angle is 10 less than three times its complement. what is the measure of the angle?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
the supplement of an angle is 10 less than three times its complement. what is the measure of the angle?
~~~~~~~~~~~~~~~~~~~~~~~

Let x be the measure of the angle, in degrees. Then the supplement is (180-x) degrees and the complement is (90-x) degrees. Your equation is 180 - x = 3(90-x) - 10. Simplify and find x 180 - x = 270 - 3x - 10, 3x - x = 270 - 10 - 180, 2x = 80 x = 80/2 = 40. ANSWER. The measure of the angle is 40 degrees.

Solved.

-------------------

The solution and the answer in the post by @lynnlo both are incorrect,
so ignore his post.

Question 1210486: In triangle, R, S, T△RST, XX is the intersection of the three medians. If R, W, equals, 46RW=46, find W, TWT.
R
W
T
V
S
U
X
Answer
W, T, equalsWT=

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
This problem relies on the fundamental property of the **medians** of a triangle, which is that they intersect at a single point called the **centroid** ($X$).
In $\triangle RST$:
* $RX$, $SX$, and $TX$ are medians.
* $W$ is the midpoint of the side $ST$. This means the segment $RW$ is the median drawn from vertex $R$ to side $ST$.
* $X$ is the centroid, located on the median $RW$.
### Centroid Theorem
The Centroid Theorem states that the centroid divides each median into two segments with a ratio of **2:1**. The segment connecting the vertex to the centroid is twice as long as the segment connecting the centroid to the midpoint of the opposite side.
In this case, for the median $RW$:
$$RX = 2 \cdot XW$$
### Solving the Problem
You are given the length of the entire median, $RW = 46$.
The entire median length is the sum of its two segments:
$$RW = RX + XW$$
Since $RX = 2 \cdot XW$, we can substitute this into the equation:
$$RW = 2 \cdot XW + XW$$
$$RW = 3 \cdot XW$$
Substitute the given value $RW=46$:
$$46 = 3 \cdot XW$$
$$XW = \frac{46}{3}$$
The problem asks for the length of **$WT$**.
***Wait!*** Looking carefully at the diagram and the given information:
* The point $W$ is the midpoint of $ST$. Thus $RW$ is a median.
* The segment $WT$ is **half of the side $ST$**, since $W$ is the midpoint of $ST$.
The length of the median $RW$ ($46$) tells us the lengths of the segments $RX$ and $XW$, but it **does not directly** give us the length of $WT$.
**Conclusion based on standard geometry interpretation:**
Since the problem gives the length of the median $RW$ and asks for the length of the segment $WT$, there must be a typographical or labeling error in the problem statement or the question being asked.
* If the question meant to ask for **$XW$** (the segment of the median): $XW = \mathbf{46/3}$.
* If the question meant to ask for **$RX$** (the segment of the median): $RX = 2(46/3) = \mathbf{92/3}$.
* If the question meant to give **$ST$** ($ST=46$) and ask for $WT$: $WT = 46/2 = 23$.
**Assuming the intended question was to relate the segments of the median $RW$:**
The length of $WT$ cannot be determined from the length of $RW$. However, $W$ is shown as the midpoint of $ST$, so $\mathbf{WT = \frac{1}{2} ST}$.
Given the context of centroid problems, it is highly likely the intention was to ask for **$XW$**.
$$XW = 46 / 3 \approx 15.33$$
Since I must answer the question as stated, and $WT$ is unrelated to $RW$ (except that $W$ is on $ST$), I cannot solve it. However, if the labels in the diagram meant to imply $W$ is the midpoint of $RT$, then $WT$ would be the other segment of the median from $S$.
**Let's assume the question meant to ask for the length of the segment $\mathbf{XW}$**:
$$XW = \mathbf{46/3}$$

Question 732951: what expression represents the sum of the first n multiples of 15 ?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
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what expression represents the sum of the first n multiples of 15 ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

These multiples form arithmetic progression N(k) = 15k, k = 1, 2, 3, . . . , n. First term is 15, the common difference is 15. The sum of the first n terms is 15*(1 + 2 + 3 + . . . + n) = . ANSWER

Solved.

Question 732961: what expression represents the first n multiples of 15 ?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
what expression represents the first n multiples of 15 ?
~~~~~~~~~~~~~~~~~~~~~~~~

N(k) = 15k, k = 1, 2, 3, . . . , 15. ANSWER

Solved.

Question 733073: What is the supplement of a' 79 degree angle

Answer by ikleyn(53748)   (Show Source):
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What is the supplement of a' 79 degree angle
~~~~~~~~~~~~~~~~~~

The supplement is 180 - 79 = 101 degrees.

Question 744577: please help me solve this question
if radius= 10 cm, diameter= 20 cm and total surface area= 1540 sq.cm then height= ?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

If the radius is given, there is no need specify the diameter.

What is really necessary, is to specify the shape of a solid.

Question 503883: I need help setting up the following problem.
The measure of angle A is half the difference between its complement and supplement. Find the measure of angle A.
Thank you!
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!

Does this make any sense?
complement, 45.
supplement, 135.
Original statement,

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
The measure of angle A is half the difference between its complement and supplement.
Find the measure of angle A.
~~~~~~~~~~~~~~~~~~~~~~~~~~

The measure of the complement angle is 90-A degrees. The measure of the supplement angle is 180-A degrees. The difference between the supplement angle measure and the complement angle measure is (180-A) - (90-A) = 90 degrees. Half of this difference is 45 degrees. Hence, the measure of angle A is 90/2 = 45 degrees.

Solved.

Question 1209965: How do i establish a 52degree angle of of a baseline?
Painting lines in a parking lot and they need to be 52 degee angle from the curbline
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Establishing a precise 52-degree angle for parking lot lines can be done with a few methods, ranging from simple to more advanced. Here's a breakdown:
**1. Simple Protractor Method (Less Precise, Good for Small Areas):**
* **Tools:**
* Large protractor or angle finder
* Measuring tape
* Chalk or marking paint
* **Steps:**
1. **Establish Baseline:** Accurately mark your curbline as the baseline.
2. **Mark a Point:** Choose a starting point along the curbline.
3. **Use Protractor:** Place the center of the protractor at the marked point, aligning the 0-degree line with the curbline.
4. **Mark the Angle:** Find the 52-degree mark on the protractor and make a small mark on the ground.
5. **Create Line:** Use the marked point and the 52-degree mark to draw a line with chalk or marking paint.
6. **Repeat:** Repeat this process at various points along the curbline to create parallel lines.
* **Limitations:** This method is prone to errors, especially over longer distances. Small inaccuracies in the protractor placement will accumulate.
**2. Trigonometry and Measuring (More Precise):**
* **Tools:**
* Measuring tape (long)
* Calculator (with trigonometric functions)
* Chalk or marking paint
* String line or straight edge
* **Steps:**
1. **Establish Baseline:** Accurately mark your curbline.
2. **Mark a Point:** Choose a starting point along the curbline.
3. **Measure a Distance:** Measure a known distance along the curbline from the starting point (e.g., 10 feet).
4. **Calculate Offset:** Use the tangent function to calculate the offset distance.
* `offset = distance * tan(52 degrees)`
* tan(52) is about 1.2799. So for every 10 foot of baseline, the offset is about 12.8 feet.
5. **Mark Offset:** Measure the calculated offset distance perpendicular to the curbline and mark the point.
6. **Create Line:** Connect the starting point on the curbline with the offset point to create the 52-degree line.
7. **Repeat:** Repeat this process at various points along the curbline, ensuring the offset measurements are consistently perpendicular.
* **Advantages:** This method is more accurate than the protractor method, especially for longer lines.
**3. Using a Transit or Theodolite (Highly Precise):**
* **Tools:**
* Transit or theodolite (surveying equipment)
* Tripod
* Measuring tape
* Chalk or marking paint
* **Steps:**
1. **Set Up Transit:** Set up the transit on a tripod at a starting point.
2. **Establish Baseline:** Align the transit with the curbline.
3. **Set Angle:** Set the transit to 52 degrees.
4. **Mark Points:** Use the transit's line of sight to mark points at intervals along the 52-degree line.
5. **Create Line:** Connect the marked points with chalk or marking paint.
6. **Repeat:** Move the transit and repeat the process for other lines.
* **Advantages:** This method is the most accurate, as transits and theodolites are designed for precise angle measurements.
**4. Using a Construction Laser with Angle Capabilities (Precise and Efficient):**
* **Tools:**
* Construction laser with angle setting capabilities.
* Tripod.
* Measuring tape.
* Marking paint.
* **Steps:**
1. **Set up laser:** Set up laser on tripod.
2. **Establish Baseline:** Align the laser with the curbline.
3. **Set Angle:** Set the laser to 52 degrees.
4. **Mark Lines:** The laser will project a line at the correct angle, which you can then mark.
5. **Repeat:** Move the laser as needed.
* **Advantages:** Very accurate and fast.
**Important Considerations:**
* **Accuracy:** For parking lot lines, a moderate level of accuracy is usually sufficient. However, for critical applications, use a transit or laser.
* **Safety:** Wear appropriate safety gear, such as safety glasses and high-visibility clothing.
* **Weather:** Wind and rain can affect accuracy, especially when using string lines or lasers.
* **Local Regulations:** Check local regulations for parking lot line specifications.
For most parking lot applications the trigonometry method or a construction laser will work very well.

Question 1186313: A triangular lot ABC with BC=400m and B=50° is divided into two parts by the line DE=150m which is parallel to BC. Point D and E are located on the side of AB and AC, respectively. The area of the segment BCED is 50,977 sq.m.
a. Find the angle that the side AC makes with the side BC
b. Find the area of ABC
c. Find the area of ADE

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Here's how to solve this problem:
**a. Find angle C (the angle that AC makes with BC):**
1. **Area of BCED:** We know the area of trapezoid BCED is 50,977 sq.m. The formula for the area of a trapezoid is (1/2) * (base1 + base2) * height. In our case, the bases are BC and DE, and the height is the perpendicular distance between them. However, we don't know the height yet.
2. **Triangles ADE and ABC are similar:** Since DE is parallel to BC, triangles ADE and ABC are similar. This means their corresponding angles are equal, and the ratio of their corresponding sides is constant.
3. **Ratio of corresponding sides:** DE/BC = 150/400 = 3/8. Let this ratio be k.
4. **Ratio of areas:** The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. Area(ADE)/Area(ABC) = k² = (3/8)² = 9/64
5. **Area of ABC:** Let Area(ABC) = X. Then Area(ADE) = (9/64)X. Also, Area(BCED) = Area(ABC) - Area(ADE) = X - (9/64)X = (55/64)X. We know that the area of BCED is 50977. So (55/64)X = 50977, from which X = 59200.
6. **Area of ADE:** Area(ADE) = (9/64) * 59200 = 8325
7. **Area of ABC (using another approach):** Area(BCED) = Area(ABC) - Area(ADE). 50977 = X - (9/64)X. So, X = 59200.
8. **Area of ABC (using sine formula):** Area(ABC) = (1/2) * BC * AB * sin(B) = (1/2) * BC * AC * sin(C). We know BC = 400 and B = 50°. We can express AB and AC in terms of BC using the Law of Sines:
AB/sin(C) = BC/sin(A) and AC/sin(B) = BC/sin(A)
9. **Angle A:** Since the sum of the angles in a triangle is 180, we have A + B + C = 180, or A + 50 + C = 180. So A = 130 - C.
10. **Area of ABC (substituting):** 59200 = (1/2) * 400 * AC * sin(C). So AC = 296/sin(C)
11. **Law of Sines:** AC/sin(50) = 400/sin(130-C). 296/sin(C) * sin(50) = 400/sin(130-C). 0.5662/sin(C) = 400/(sin130cosC - cos130sinC). 0.5662(sin130cosC - cos130sinC) = 400sinC. 0.5662(0.766cosC + 0.6428sinC) = 400sinC. 0.4349cosC + 0.364sinC = 400sinC. 0.4349cosC = 399.636sinC. tanC = 0.4349/399.636. C = 0.0623 degrees. It seems that this is wrong.
12. **Let's take a different approach**: Area of ABC = (1/2) * BC * h where h is perpendicular from A to BC. 59200 = (1/2)*400*h. h = 296. h = ABsinB = ACsinC. AB/AC = sinC/sinB. AB = ACsinC/sin50. Area of ABC = (1/2)BC*ACsinC = (1/2)*400*ACsinC = 59200. 200ACsinC = 59200. ACsinC = 296. Area of ADE = (1/2)*DE*h1 = (1/2)*150*h1 = 8325. h1 = 111. h1/h = 150/400. h1 = (150/400)*h = 0.375*296 = 111.
13. **Use the given area of BCED**: Area(BCED) = (1/2)(BC+DE)h2. 50977 = (1/2)(400+150)h2. h2 = 185.37
14. h2 is the height of the trapezoid which is not the height of the triangle.
**b. Find the area of ABC:**
We have already found the area of ABC = 59200 sq.m.
**c. Find the area of ADE:**
We have already found the area of ADE = 8325 sq.m.
**a. Angle C**
Area(ABC) = (1/2) * BC * AC * sin(C)
59200 = (1/2) * 400 * AC * sin(C)
296 = AC * sin(C)
We also know that Area(ADE) = (1/2) * DE * AE * sin(A) and since triangles are similar, the angles are the same.
Since we have two unknowns (AC and C), we need another equation.
We know that the ratio of the sides is 150/400 = 3/8. So AE/AC = AD/AB = DE/BC = 3/8.
AD/AB = 3/8, so AB = (8/3)AD. Also AE/AC = 3/8, so AC = (8/3)AE.
Area(ABC) = (1/2) * BC * AB * sin(B)
59200 = (1/2) * 400 * AB * sin(50)
AB = 59200/(200sin50) = 387.44m
AD = (3/8)AB = 145.29m
AE = (3/8)AC
DE/BC = AE/AC = AD/AB = 3/8
Area(ADE) = (1/2) * AD * AE * sin(A)
AC/sin50 = 400/sinA
AC = 400sin50/sinA
59200 = (1/2)*400*ACsinC
296 = ACsinC
ACsinC = 296
AC = 296/sinC
296/sinC * sin50 = 400/sinA
sinA = (400sinCsin50)/296
A = 180 - 50 - C
sin(130-C) = (400sinCsin50)/296
sin130cosC - cos130sinC = 2.28sinC
0.766cosC + 0.6428sinC = 2.28sinC
0.766cosC = 1.6372sinC
tanC = 0.766/1.6372 = 0.467
C = 25.02 degrees.
**Final Answers:**
a. Angle C ≈ 25.02 degrees
b. Area of ABC = 59200 sq.m.
c. Area of ADE = 8325 sq.m.

Question 1191303: Complete the formal proof of the following theorem.
The bisectors of two adjacent supplementary angles form a right angle.
There is a line and three rays. There are six labeled points and four labeled angles.
Line A D is horizontal. B is on this line and between A and D.
Ray B C goes up and to the right.
Ray B F goes up and to the right. Point F is below and to the right of point C.
Ray B E goes up and to the left.
Angle A B E is labeled 1 and is marked with one arc.
Angle C B E is labeled 2 and is marked with one arc.
Angle C B F is labeled 3 and is marked with two arcs.
Angle D B F is labeled 4 and is marked with two arcs.
Given:
∠ABC is supplementary to ∠CBD.
BE bisects ∠ABC.
BF bisects ∠CBD.
Prove: ∠EBF is a right angle.
Statements Reasons
1.
---Select---
1. Given
2.
---Select---
2. The sum of the measures of supplementary ∠s is 180°.
3. m∠ABC = m∠1 + m∠2;
---Select---
3.
---Select---
4.
---Select---
4. Substitution
5.
BE bisects ∠ABC;

BF bisects ∠CBD.
5.
---Select---
6.
---Select---
6. If a ray bisects an ∠, then two ∠s of = measure are formed.
7. m∠2 + m∠2 + m∠3 + m∠3 = 180° 7.
---Select---
8.
2 · m∠2 + 2 · m∠3 = 180°
8. Combine like terms.
9. m∠2 + m∠3 = 90° 9.
---Select---
10.
---Select---
10. Angle-Addition Postulate
11.
---Select---
11. Substitution
12.
---Select---
12.
---Select---

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Here's the completed two-column proof:
**Given:**
∠ABC is supplementary to ∠CBD.
BE bisects ∠ABC.
BF bisects ∠CBD.
**Prove:**
∠EBF is a right angle.
| Statements | Reasons |
|---|---|
| 1. ∠ABC is supplementary to ∠CBD. | 1. Given |
| 2. m∠ABC + m∠CBD = 180° | 2. The sum of the measures of supplementary ∠s is 180°. |
| 3. m∠ABC = m∠1 + m∠2; m∠CBD = m∠3 + m∠4 | 3. Angle-Addition Postulate |
| 4. m∠1 + m∠2 + m∠3 + m∠4 = 180° | 4. Substitution |
| 5. BE bisects ∠ABC; BF bisects ∠CBD. | 5. Given |
| 6. m∠1 = m∠2; m∠3 = m∠4 | 6. If a ray bisects an ∠, then two ∠s of = measure are formed. |
| 7. m∠2 + m∠2 + m∠3 + m∠3 = 180° | 7. Substitution |
| 8. 2 · m∠2 + 2 · m∠3 = 180° | 8. Combine like terms. |
| 9. m∠2 + m∠3 = 90° | 9. Division Property of Equality |
| 10. m∠EBF = m∠2 + m∠3 | 10. Angle-Addition Postulate |
| 11. m∠EBF = 90° | 11. Substitution |
| 12. ∠EBF is a right angle. | 12. Definition of a right angle |

Question 1209519: (38) Find the value of w.
Link to diagram: https://ibb.co/202BWZT9
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
In a square, the diagonal divides the square into two right-angled triangles. The sides of the square form the legs of the right triangle, and the diagonal is the hypotenuse.
We can use the Pythagorean theorem to find the length of the diagonal (w). The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a² + b² = c²
In this case, the sides of the square are a = 10 and b = 10, and the diagonal is c = w. So:
10² + 10² = w²
100 + 100 = w²
200 = w²
To find w, we take the square root of both sides:
w = √200
w = √(100 * 2)
w = 10√2
Therefore, the length of the diagonal (w) is 10√2.

Question 1191305: Complete the formal proof of the theorem.
If two line segments are congruent, then their midpoints separate these segments into four congruent segments.
Line segment A B is above line segment D C. The two segments appear to be the same length. Points M and N are the midpoints of segments A B and D C, respectively.
Given:
AB ≅ DC
M is the midpoint of AB.
N is the midpoint of DC.
Prove:
AM ≅ MB ≅ DN ≅ NC
Statements Reasons
1.
AB ≅ DC
1.
Given

Correct: Your answer is correct.
2.
AB = DC

Correct: Your answer is correct.
2. If segments are ≅, then they are = in length.
3. AB = AM + MB;
AM + MB = DN + NC

Incorrect: Your answer is incorrect.
3.
Given

Incorrect: Your answer is incorrect.
4.
AM = MB = DN = NC

Incorrect: Your answer is incorrect.
4. Substitution
5.
M is the midpoint of AB;

N is the midpoint of DC.
5.
Given

Correct: Your answer is correct.
6.
AM = MB and DN = NC

Correct: Your answer is correct.
6. The midpoint of a segment forms two segments = in measure.
7. AM + AM = DN + DN 7.
Substitution

Correct: Your answer is correct.
8.
2 · AM = 2 · DN
8. Combine like terms.
9. AM = DN 9.
Division Property of Equality

Correct: Your answer is correct.
10.
AM + MB = DN + NC

Incorrect: Your answer is incorrect.
10. Substitution
11.
AM ≅ MB ≅ DN ≅ NC
11.
If segments are = in length, then they are ≅.

Correct: Your answer is correct.

Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

This request is about a SELF-EVIDENT statement.

Only out of boredom can one engage in such proofs.

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Here's the completed proof with the corrected steps:
**Given:**
AB ≅ DC
M is the midpoint of AB.
N is the midpoint of DC.
**Prove:**
AM ≅ MB ≅ DN ≅ NC
| Statements | Reasons |
|---|---|
| 1. AB ≅ DC | 1. Given |
| 2. AB = DC | 2. If segments are ≅, then they are = in length. |
| 3. AM + MB = AB; DN + NC = DC | 3. Segment Addition Postulate |
| 4. AM + MB = DN + NC | 4. Substitution (since AB=DC) |
| 5. M is the midpoint of AB; N is the midpoint of DC. | 5. Given |
| 6. AM = MB and DN = NC | 6. The midpoint of a segment forms two segments = in measure. |
| 7. AM + AM = DN + DN | 7. Substitution (substituting MB for AM and NC for DN in step 4) |
| 8. 2 · AM = 2 · DN | 8. Combine like terms. |
| 9. AM = DN | 9. Division Property of Equality |
| 10. AM = MB and DN = NC (from step 6) and AM = DN (from step 9), therefore AM = MB = DN = NC | 10. Transitive Property of Equality |
| 11. AM ≅ MB ≅ DN ≅ NC | 11. If segments are = in length, then they are ≅. |

Question 1191300: Use the drawing in which
AC intersects DB
at point O to answer the question.
Two lines intersect at point O to form four angles.
Line A C goes from point A in the top left, through point O in the center, to point C in the bottom right.
Line D B goes from point D in the bottom left, through O on line A C, to point B in the top right.
The angle at the bottom, ∠C O D, is labeled 1.
The angle on the left, ∠D O A, is labeled 2.
The angle at the top, ∠A O B, is labeled 3.
The angle on the right, ∠B O C, is labeled 4.
If
m∠2 =
x
2
− 30
°

and
m∠3 =
x
3
+ 60
°

,
find x and m∠2 in degrees.
x =
m∠2 =
°
Need Help? Read It
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(53748)   (Show Source):
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.

The post is in unreadable format.

In this format, it only good to throw it to the closest garbage bin.

Copy-paste does not work at this forum.

If a problem includes formulas, expressions, equations, you should print it manually
using your keyboard. Then it will work.

Answer by CPhill(2189)   (Show Source):
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Here's how to solve for x and m∠2:
1. **Recognize the relationship between the angles:** ∠2 and ∠3 are vertical angles. Vertical angles are congruent, meaning they have the same measure.
2. **Set up an equation:** Since m∠2 = m∠3, we can write:
x² - 30 = x³ + 60
3. **Rearrange the equation:** To solve for x, we need to rearrange the equation. However, this equation involves x² and x³, which makes it a cubic equation, and isn't easy to solve directly. The problem likely intended a linear relationship. It is possible there was a typo in the original problem. If we assume that the intended relationship was linear rather than polynomial, and assuming that the intended measures were m∠2 = x - 30 and m∠3 = x + 60, then we can proceed as follows:
x - 30 = x + 60
4. **Solve for x:**
Subtract x from both sides:
-30 = 60 This is not possible.
It's possible that there was a typo in the original problem. If the angle measures were instead m∠2 = x - 30 and m∠3 = 2x + 60, then we would proceed as follows:
x - 30 = 2x + 60
-90 = x
5. **Substitute x back into the equation for m∠2:**
m∠2 = x - 30
m∠2 = -90 - 30
m∠2 = -120°
Since angle measures can't be negative, it's likely there was a typo in the original problem. Double-check the given angle measures. If they were m∠2 = x-30 and m∠3 = x+60, then there is no solution. If they were m∠2 = x-30 and m∠3 = 2x+60, then x = -90 and m∠2 = -120.
**If you can provide the correct angle measures, I can help you solve the problem accurately.**

Question 1209511: bd 5x 26 bc=2x+1 and ac=43 find ab
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
incomplete and incomprehensible

Question 1209478: In triangle ABC, DE is parallel to BC and AD = DE = BE.
If BE is the angle bisector of angle CBA, then find the measure of angle ACB.
Link to the graph: https://ibb.co/h9RJzZR
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Since BE bisects angle CBA, let x be the measure of each of angles CBE and EBD.

In triangle DEB, DE = BE, so the measure of angle BDE is also x.

That makes the measure of angle EDA 180-x.

But DE and BC are parallel, so angles EDA and CBA are congruent.

2x = 180-x
3x = 180
x = 60

In triangle EDA, AD = DE, so angles DAE and DEA are congruent. Then, since the measure of angle EDA is 180-x=120, the measure of each of angles DAE and DEA is x/2 = 30.

And again DE and BC are parallel, so the measure of angle ACB is also 30.

ANSWER: 30 degrees

Question 1209480: The center of the circle is O, angle TQR = 46 degrees and angle QTO = 51 degrees.
Find the measure of angle RPO.
Link to graph: https://ibb.co/bmnmVgb
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

The measure of angle TQR is given as 46 degrees, so the measure of angle TQP is 180-46 = 134 degrees.

That means the measure of major arc PT is 2*134 = 268 degrees.

That in turn means the measure of minor arc PT is 360-368 = 92 degrees.

And since central angle POT cuts off minor arc PT, the measure of angle POT is 92 degrees.

The measure of angle QTO was given as 51 degrees.

Now we have the measures of three of the four angles in quadrilateral POTQ, so the measure of angle RPO (= measure of angle QPO) is 360-(134+51+92) = 83 degrees.

ANSWER: 83 degrees

Question 1209487: (31) If AB = BC = CD and ∠BCA = 54°, then find the measure of angle ABD.
Diagram: https://ibb.co/zVZdT58
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
(31) If AB = BC = CD and ∠BCA = 54°, then find the measure of angle ABD.
Diagram: https://ibb.co/zVZdT58
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In triangle ABC, AB = BC (given). It means that triangle ABC is isisceles, with base angles BAC and ACB of 54 degrees. Hence, the angle ABC at the vertex is 180 - 54 - 54 = 180 - 108 = 72 degrees. In triangle BCD, BC = CD (given). It means that triangle BCD is isosceles. Hence, its angles CBD and CDB are congruent, as the base angles. The sum of these angles, CBD and CDB, is equal to the exterior angle ACD, which is 54 degrees. Hence, angle CBD is half of 54 degrees, i.e. 27 degrees. Now, angle ABD is the sum of angles ABC (72 degrees) and CBD (27 degrees) So, angle ABD = 72 + 27 = 99 degrees. At this point, the problem is solved completely. ANSWER. Angle ABD is 99 degrees.

Solved.

Question 1209462: An angle measures 68° more than the measure of its supplementary angle. What is the measure of each angle
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Formally....

Let x be the measure of the angle
Then 180-x is the measure of its supplement

The angle measure is 68 degrees more than the measure of the supplement:

x = (180-x)+68
x = 248-x
2x = 248
x = 124

ANSWERS:
the angle: x = 124 degrees
the supplement: 180-x = 180-124 = 56 degrees

Informally....

The sum of the measures of an angle and its supplement is 180 degrees.

That means the measures of the two angles differ from 90 degrees by equal amounts.

Since in this problem the difference between the measurements of the two angles is 68 degrees, the measures of the two angles differ from 90 degrees by 68/2 = 34 degrees.

So the measures of the two angles are 90+34 = 124 degrees and 90-34 = 56 degrees.

ANSWERS: 124 degrees and 56 degrees

Question 1209444: Verify the triangles are similar using AA theorem. Show your work
What type of angles are angle P?
What two angle pairs are congruent
______ & ______
______ & ______
https://ibb.co/f98nbmr
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Compute acute angles in right-angles triangles.

After computing, compare their measures and make your conclusion.

This problem is of the children's level of difficulty.

It's indecent to come with such questions.

When you come with such questions, you demonstrate that you do not want to make
any single movement by your finger and any movement by your cerebral convolution
to solve the problem on your own.

        It is a SHAME !

Question 1186931: An isosceles triangle ABC, in which AB = BC = 6√2 and AC = 12 is folded along the altitude BD so that planes ABD and BDC form a right dihedral angle. Find the angle between side AB and its new
position.

Found 2 solutions by ikleyn, yurtman:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
An isosceles triangle ABC, in which AB = BC = 6√2 and AC = 12 is folded along the altitude BD
so that planes ABD and BDC form a right dihedral angle. Find the angle between side AB and its
new position.
~~~~~~~~~~~~~~~~~~~~

        The solution by the other tutor (which, I believe, is an AI solution) is FATALLY WRONG.
        I came to bring a correct solution.

In the isosceles triangle ABC AB^2 + BC^2 = + = 36*2 + 36*2 = 72 + 72 = 144 and AC^2 = = 144. It means that triangle ABC is a right-angled isosceles triangle with the right angle at B, hypotenuse BC and the legs AB and BC. Now imagine that the triangle ABC is folded along the altitude BD so the planes ABD and BDC form a right dihedral angle. Let plane BDC remains in its original position and vertex A takes new position A' in space. Then the segments A'D and CD form the right angle. Consider triangle A'DC. It is a right-angled triangle with the right angle D and the legs A'D and CD. These legs have equal length A'D = CD = 12/2 = 6, A'D= AD and AD and CD are equal halves of the base AC of the triangle ABC. Therefore, the hypotenuse A'C of the triangle ADC has the length = = . Thus the triangle A'BC (after folding) has congruent sides A'B, A'C and BC: the length of each of these sides is . Thus the triangle A'BC is an equilateral triangle and all its angles have the measure of 60°. In particular, the angle A'BC is 60°. The angle between AB and its new position A'B is the same 60°. The proof is the same.

Solved.

Answer by yurtman(42)   (Show Source):
You can put this solution on YOUR website!
**1. Find the Height (Altitude) of the Isosceles Triangle**
* **Use the Pythagorean Theorem:**
* In right triangle ABD (where D is the midpoint of AC):
* AB² = AD² + BD²
* (6√2)² = AD² + 6²
* 72 = AD² + 36
* AD² = 36
* AD = 6
**2. Visualize the Folding**
* When triangle ABC is folded along altitude BD, side AB rotates around point B.
* Imagine the original position of AB and its new position after folding. This creates a dihedral angle (the angle between the two planes).
**3. Determine the Angle Between AB and its New Position**
* The angle between AB and its new position is twice the angle between AB and the plane BDC.
* Let's call this angle θ.
* **Consider right triangle ABD:**
* tan(θ) = AD / BD = 6 / 6 = 1
* θ = arctan(1) = 45°
* **Angle between AB and its new position:** 2 * θ = 2 * 45° = 90°
**Therefore, the angle between side AB and its new position after folding is 90 degrees.**

Question 1209183: Angle T and Angle S are supplementary. Angle T measures (5x-3)
and Angle S measures (2x+1)
.
Find the measure of each angle.
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Supplementary angles sum up to 180 degrees T + S = 180 degrees, (5x-3) + (2x+1) = 180 7x - 2 = 180 7x = 180 + 2 7x = 182 x = 182/7 = 26 T = 5x-3 = 5*26-3 = 127 degrees, S = 2x+1 = 2*26+1 = 53 degrees.

Solved.

Question 1208920: In the diagram, find the measure of ∠ BFD.
https://ibb.co/xsSFXH4
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Consider quadrilateral DECF. As for any quadrilateral, the sum of its interior angles is 360°. The angles D and C of this quadrilaterals are right angles; their sum is 90° + 90° = 180°. Hence, the sum of angles F and E of this quadrilateral is 360° - 180° = 180°. Thus the angle CFD is the supplementary angle to angle E: ∠CFD = 180° - 57° = 123°. But the angle BFD is the supplementary angle to the angle CFD. So, we conclude that ∠BFD = 180° - 123° = 57°. ANSWER. ∠BFD = 57°.

Solved.

Question 1208657: the supplement of an angle is four times the complement of the angle find the measure of half the supplement of the angle

Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

x = the unknown angle
180-x = the supplement of this angle
90-x = the complement of the original angle
Supplementary angles add to 180 degrees.
Complementary angles add to 90 degrees.

The instructions mention that the supplement is 4 times the complement,
supplement = 4*complement
180-x = 4*(90-x)
180-x = 360-4x
-x+4x = 360-180
3x = 180
x = 180/3
x = 60 is the original angle
180-x = 180-60 = 120 is the supplement, which cuts in half to 60 degrees, so that is the final answer.

Question 1208476: An angle measures 153.2° less than the measure of its supplementary angle. What is the measure of each angle?

Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!

.
.

Question 1208246: Angles A and B complimentary angles. Find angles A and B.
Angle A = 4x+31
Angle B =-2x+44
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Angles A and B are complementary angles. Find angles A and B.
Angle A = 4x+31
Angle B =-2x+44
~~~~~~~~~~~~~~~~~~~~~~

Complementary angles sum up to 90 degrees (4x+31) + (-2x+44) = 90 degrees 4x + 31 - 2x + 44 = 90 4x - 2x + 75 = 90 2x = 90 - 75 2x = 15 x = 15/2 = 7.5 degrees. Angle A is 4x+31 = 4*7.5+31 = 30+31 = 61 degrees. Angle B is -2x+44 = -2*7.5 + 44 = -15 + 44 = 29 degrees. CHECK. 61 + 29 = 90 degrees. ! correct !

Solved.

-------------

To learn about complementary angles, read the lessons
    - Angles basics
    - Vertical angles
    - HOW TO solve problems on supplementary, complementary or vertical angles - Examples
    - Solved problems on supplementary and complementary angles
in this site.

Consider these lessons as your handbook, textbook, guide, tutorials and (free of charge) home teacher.

You will find there many similar problems, solved with complete explanations.

Learn the subject from there.

Question 1208163: the supplement of an angle is 16 less than 3 times the angle, find the angle

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
the supplement of an angle is 16 less than 3 times the angle, find the angle
~~~~~~~~~~~~~~~~~~~~~

Let x be the angle (in degrees). Then the supplement to this angle is 180-x degrees, and we write the given part as an equation 180-x = 3x - 16. Then we simplify and solve this equation 180 + 16 = 3x + x 196 = 4x x = 196/4 x = 49 degrees. ANSWER

Solved.

---------------------

To learn about supplementary angles, read the lessons
    - Angles basics
    - Vertical angles
    - HOW TO solve problems on supplementary, complementary or vertical angles - Examples
    - Solved problems on supplementary and complementary angles
in this site.

You will find there many similar problems, solved with complete explanations.

Consider these lessons as your handbook, textbook, guide, tutorials and (free of charge) home teacher.

Learn the subject from there.

Question 1207551: What would the three legs be for a 125.25 degree angle
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Makes no sense....

Since there are three legs, we can assume you are talking about triangles.

If that is the case, then the two legs forming the 125.25 degree angle can be any lengths. For each pair of lengths for those legs, the length of the other side will be uniquely determined.

But there is no single "answer" to the question.

Question 1207028: The measure of an angle is 139.8°. What is the measure of its supplementary angle?
Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website!

The measure of an angle is 139.8°. What is the measure of its supplementary angle?
Supplementary angles are the angles that add up to 180 degrees.

180-139.8 = 40.2 degrees
The measure of the supplementary angle is 40.2 degrees

Question 1206694: line segment CE has two bisectors. the first line OR is a perpendicular bisector. what can you conclude about the second bisector, line ST?
A. ST is equal to OR
B. ST is perpendicular to CE
C. ST is equal to CE
D.ST is not perpendicular to CE
E.ST is perpendicular to OR
Found 2 solutions by greenestamps, MathLover1:
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

The two lines OR and ST both bisect segment CE, so they both pass through the midpoint of CE.

The statement of the problem implies that OR and ST are not the same line; so if OR is a perpendicular bisector of CE then ST is not perpendicular to CE.

ANSWER: D

Answer by MathLover1(20855)   (Show Source):
You can put this solution on YOUR website!

as you can see, answer is:
D. is not perpendicular to

Question 1206536: triangle EBT is similar to SDA. if the perimeter of EBT is 63, the perimeter of SDA is 21. BT=6x, and DA=x+3, find BT and DA
Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

The linear scale factor from EBT to SDA is 1/3 because 21/63 = 1/3.
The side lengths of SDA are 1/3 as long compared to the side lengths of EBT.

DA/BT = 1/3
(x+3)/(6x) = 1/3
3(x+3) = 6x*1
3x+9 = 6x
9 = 6x-3x
9 = 3x
x = 9/3
x = 3

BT = x+3 = 3+3 = 6
DA = 6x = 6*3 = 18

Question 1206535: assume triangle abc is similar to fgh with median ap and fq to both sides bc and gh respectively, ab=9 and fg=6 if ap is 4 greater than fq find both medians

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

ab and fg are corresponding sides, so the ratio of similarity is 9:6 = 3:2.

The lengths of the medians are linear measurements, so the ratio ap:fq = 3:2. Since ap is 4 greater than fq....

ANSWERS: ap=12; fq=8

Question 1206495: In the figure, angle D measures 45° and angle E measures 87°. What is the measurement of angle H?
A.
42°
B.
45°
C.
48°
D.
93°
Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website!
In a triangle the sum of the angles =180 degrees
In Triangle DEF
angle D measures 45° and angle E measures 87°
angle D +angle E +angle F = 180deg
45+87+angle F = 180deg
Angle F = 180-45-87 = 48degrees
C : 48 deg

Question 1206493: An angle measures 155.2° less than the measure of its supplementary angle. What is the measure of each angle?

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

x = measure of the given angle
180-x = measure of its supplementary angle

The angle measure is 155.2 degrees less than the measure of its supplement:

x = (180-x)-155.2
x = 24.8-x
2x = 24.8
x = 12.4

ANSWERS:
angle: 12.4 degrees
supplement: 180-12.4 = 167.6