Geometry Midterm Review
Section 4-5: Isosceles and Equilateral Triangles
Isosceles Triangle Vocabulary
legs- the two congruent sides of an isosceles triangle
base- the third side, not congruent to either of the other sides
vertex angle- the angle formed by the two congruent sides
base angles- the two congruent angles, each formed by the base and a leg
Theorems
Theorem 4-3: Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent
Theorem 4-4: Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite the angles are congruent
Corollary to Theorem 4-3
If a triangle is equilateral, then the triangle is equiangular
Corollary to Theorem 4-4
If a triangle is equiangular, then the triangle is equilateral
Theorem 4-5
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base
Applying the Theorems
Explain why angle WVS is congruent to angle S
Side WV is congruent to side WS. Therefore, using the Isosceles Triangle Theorem, you
know that angle WVS is congruent to angle S
Explain why side TR is congruent to TS
Angle R is congruent to angle WVS, and angle WVS is congruent to angle S. By the
transitive property, angle R is congruent to angle S. Using the Converse of the Isosceles
Triangle Theorem, you can conclude that side TR is congruent to side TS
Find the value of y
By Theorem 4-5, side MO is perpendicular to LN, so angle MON is 90 degrees. Since
triangle LMN is an isosceles triangle, angle L is congruent to angle N. That means that
angle N is 63 degrees. The sum of all angles in a triangle is 180 degrees. Therefore,
63+90+y=180 degrees. Solve for y, and you can figure out that y equals 27 degrees.
Practice Problems
Answers to Practice Problems
1) Side VT is congruent to side VX because of the Converse of the Isosceles Triangle Theorem
2) Side UT is congruent to side UW which is congruent to side YX because of the Converse of the Isosceles Triangle Theorem
3) Side VU is congruent to VY. This is because angle X is congruent to angle UYV because they are corresponding angles. It is given in the diagram that angle X is congruent to angle VUY. By the transitive property, angle VUY is congruent to angle VYU. Using the Converse of the Isosceles Triangle Theorem, you know that side VU is congruent to VY.
4) Y=40; X=80
5) Y=35; X=110
6) Y=4; x=38