Question 175611: Hi All,
I'm a HS Senior with my last class in Geometry, I saw a problem similar to mine, but with a different fact to prove...so here it is:
Prove:If an isosceles triangle has an altitude fron the vertex to the base, then the altitude bisects the vertex angle
Given:(triangle)ABC is isosceles; (line) CD is the altitude to base (line) AB
To Prove: (line)CD bisects (angle) ACB
Plan: (this space is blank, what should I write here? It's slightly confusing)
What I do know now is that two sides are congruent, and the altitude creates two right angles. This is all pretty confusing to me, math isn't my strong suit.
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! I'm assuming that the drawing looks something like this:
Here's a Two Column Proof:
Statement Reason
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1. Triangle ABC is isosceles Given
2. Angle DAC = Angle DBC Definition of Isosceles
3. Segment CD = Segment CD Reflexive Property of Congruence
4. Angle ADC is a right angle Definition of Altitude
5. Angle BDC is a right angle Definition of Altitude
6. Angle ADC = Angle BDC All right angles are equal
7. Triangle ACD = Triangle BCD AAS Property of Congruence
8. Angle ACD = Angle BCD CPCTC
9. Segment CD bisects Angle BCD Definition of Bisected Angle
Note: CPCTC = Corresponding Parts of Congruent Triangles are Congruent.
Remember, an isosceles triangle has two equal sides and two equal base angles. Also, keep in mind that a bisected angle (one that is cut in half) has two equal parts.
So the basic course of action is:
1) Prove that triangle ACD is congruent to triangle BCD
2) From there, you can show that the corresponding parts of the triangles are congruent (in this case the top angles)
3) Since they are congruent, this means that they form a larger angle that has been bisected (a bisected angle tells us that the angle forms two smaller equal parts)
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! Prove: (line)CD bisects (angle) ACB
Plan: (this space is blank, what should I write here? It's slightly confusing)
What I do know now is that two sides are congruent, and the altitude creates two right angles
----------------------------
AC=CB----isos triangle
angle A = angle B----angles opposite equal sides
angle ADC = angle BDC ---- def of altitude
angle ABD = angle CBD------if two corresponding angles of a triangle are equal the 3rd angles are equal
CD bisects angle ACB--------definition of angle bisector
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Cheers,
Stan H.
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