Question 1191505: ▱
upper left: A
lower left: D
Upper right: B
lower right: C
Centre: E
1.) Given: AB=(3x-5) cm, BC=(2y-7) cm, CD=(x+7) cm and AD=(y+3) cm.
∠BAD measures (2a+25)∘ while ∠BCD measures (3a-15)∘.
a. what is the value of a
b. what is m∠BAD?
c. What is m∠CBA?
2.) Diagonals AC and BD meet at E. DE is 8cm and AC is 13 cm
a. How long is BD?
b. How long is AE?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this geometry problem:
**1.) Parallelogram ABCD**
Since ABCD is a parallelogram, opposite sides are equal, and opposite angles are equal.
* **a. Value of 'a':**
* ∠BAD = ∠BCD
* 2a + 25 = 3a - 15
* 40 = a
* **b. m∠BAD:**
* m∠BAD = 2a + 25
* m∠BAD = 2(40) + 25
* m∠BAD = 105°
* **c. m∠CBA:**
* Consecutive angles in a parallelogram are supplementary (add up to 180°).
* m∠BAD + m∠CBA = 180°
* 105° + m∠CBA = 180°
* m∠CBA = 75°
**Additionally, we can use the side lengths to solve for x and y:**
* AB = CD
* 3x - 5 = x + 7
* 2x = 12
* x = 6
* AD = BC
* y + 3 = 2y - 7
* 10 = y
**2.) Parallelogram ABCD with Diagonals**
Diagonals of a parallelogram bisect each other. This means they cut each other in half at the point of intersection (E).
* **a. Length of BD:**
* Since DE is 8 cm, and the diagonals bisect each other, BE is also 8 cm.
* BD = BE + DE
* BD = 8 cm + 8 cm
* BD = 16 cm
* **b. Length of AE:**
* Since AC is 13 cm, and the diagonals bisect each other, AE is half of AC.
* AE = AC / 2
* AE = 13 cm / 2
* AE = 6.5 cm
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