Section 3.7: Angle-Side Theorems
Warmup
Prove that .
Proof
| 1 | ∠D≅∠E | Given |
| 2 | seg DE≅seg DE | Reflexive property |
| 3 | △DEF≅△EDF | ASA (1,2,1) |
| 4 | seg DF &cong seg EF | CPCTC |
Homework problems
3.6 #15
Theorems
Theorems 20 and 21: two angles of a triangle are congruent if and only if (iff) their opposite sides are congruent.
The inverses of these theorems are also true. (What are those?) In fact, the length of side a is greater than the length of side b iff the angle opposite side a has greater measure than the angle opposite side b. (Proven in Chap. 15)
Theorem: equilateral vs equiangular
A triangle is equilateral iff it is equiangular
Proof of equilateral ⇒ equiangular (&lArr is similar):
| 1 | AB=AC | Given |
| 2 | ∠B≅∠C | Base angles of an isosceles triangle are congruent |
| 3 | BC=BA | Given |
| 4 | ∠C≅∠A | Base angles of an isosceles triangle are congruent |
| 5 | ∠B≅∠A | Transitive property |
| 6 | △ABC is equiangular | Def. equiangular |
Sample problem 2
Prove: the bisector of the vertex angle of an isoceles triangle is also the median to the base
First we must set up the diagram and restate in the context of the diagram:
Given: △JOM is isoceles with ∠JOM the vertex angle. Ray OK bisects ∠JOM. Prove: Seg OK is the median to the base.
Proof:
| 1 | △JOM is isoceles with ∠JOM the vertex angle. | Given |
| 2 | Seg OJ≅seg OM | Def. isosceles |
| 3 | Ray OK bisects ∠JOM | Given |
| 4 | ∠JOK≅∠KOM | Def. angle bisector |
| 5 | Seg OK≅seg OK | Reflexive property |
| 6 | △JOK≅△MOK | SAS (2,4,5) |
| 7 | Seg JK≅seg MK | CPCTC |
| 8 | Seg OK is the median to the base | Def. median |