Ch 9 6 The Converse of the Isosceles

Ch 9 -6 The Converse of the Isosceles Triangle Theorem C. N. Colon Geometry St. Barnabas HS

Introduction Isosceles triangles can be seen throughout our daily lives in structures, supports, architectural details, and even bicycle frames. Isosceles triangles are a distinct classification of triangles with unique characteristics and parts that have specific names. In this lesson, we will explore the qualities of isosceles triangles.

Key Concepts • Isosceles triangles have at least two congruent sides, called legs. • The angle created by the intersection of the legs is called the vertex angle. • Opposite the vertex angle is the base of the isosceles triangle. • Each of the remaining angles is referred to as a base angle. The intersection of one leg and the base of the isosceles triangle creates a base angle.

Key Concepts

Key Concepts Theorem Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the congruent sides are congruent.

Key Concepts • If the Isosceles Triangle Theorem is reversed, then that statement is also true. • This is known as the Converse of the Isosceles Triangle Theorem.

Key Concepts Theorem Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Key Concepts • If the vertex angle of an isosceles triangle is bisected, the bisector is perpendicular to the base, creating two right triangles. • In the diagram that follows, D is the midpoint of .

Key Concepts • Equilateral triangles are a special type of isosceles triangle, for which each side of the triangle is congruent. • If all sides of a triangle are congruent, then all angles have the same measure.

Key Concepts Theorem If a triangle is equilateral then it is equiangular, or has equal angles.

Key Concepts • Each angle of an equilateral triangle measures 60˚ • Conversely, if a triangle has equal angles, it is equilateral.

Key Concepts, continued Theorem If a triangle is equiangular, then it is equilateral.

Key Concepts • These theorems and properties can be used to solve many triangle problems.

Common Errors/Misconceptions • • incorrectly identifying parts of isosceles triangles not identifying equilateral triangles as having the same properties of isosceles triangles incorrectly setting up and solving equations to find unknown measures of triangles misidentifying or leaving out theorems, postulates, or definitions when writing proofs

YOU TRY Determine whether with vertices A (– 4, 5), B (– 1, – 4), and C (5, 2) is an isosceles triangle. If it is isosceles, name a pair of congruent angles.

Use the distance formula to calculate the length of each side. Calculate the length of .

Calculate the length of Substitute A (– 4, 5) and B (– 1, – 4) for (x 1, y 1) and (x 2, y 2). Simplify.

Calculate the length of Substitute (– 1, – 4) and (5, 2) for (x 1, y 1) and (x 2, y 2). Simplify.

Calculate the length of Substitute (– 4, 5) and (5, 2) for (x 1, y 1) and (x 2, y 2). Simplify. .

Determine if the triangle is isosceles. A triangle with at least two congruent sides is an isosceles triangle. , so is isosceles.

Identify congruent angles. If two sides of a triangle are congruent, then the angles opposite the sides are congruent. ✔

Geogebra is a graphing program that can be used to illustrate the properties of isosceles triangles.

Find the values of x and y.

Make observations about the figure. The triangle in the diagram has three congruent sides. A triangle with three congruent sides is equilateral. Equilateral triangles are also equiangular.

The measure of each angle of an equilateral triangle is 60˚. An exterior angle is also included in the diagram. The measure of an exterior angle is the supplement of the adjacent interior angle.

Determine the value of x. The measure of each angle of an equilateral triangle is 60˚. Create and solve an equation for x using this information.

Equation Solve for x. The value of x is 9.

Determine the value of y. The exterior angle is the supplement to the interior angle. The interior angle is 60˚ by the properties of equilateral triangles. The sum of the measures of an exterior angle and interior angle pair equals 180. Create and solve an equation for y using this information.

Equation Simplify. Solve for y. The value of y is 13. ✔

Using Geogebra to solve this problem:

Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent then the sides opposite to these angles are congruent.

Converse of Isosceles Triangle Theorem PROOF If two angles of a triangle are congruent, then the sides opposite to these angles are congruent. Draw , the bisector of the vertex angle Since is the angle bisector, bisector By the Reflexive Property, Property It is given that . Therefore, by AAS congruence, . Since corresponding parts of congruent triangles are congruent,

HOMEWORK p. 361# 4 -20 (mo 4)

Ch 9 -7 Proving Right Triangles Congruent by Hypotenuse Leg (HL) C. N. Colόn Geometry Reference: SIMON PEREZ.

CONGRUENT TRIANGLES A K B C ABC L M KLM by CPCTC Corresponding Parts of Congruent Triangles are Congruent We would have to prove that all six pairs of corresponding parts are congruent!

CONGRUENT TRIANGLES by Side -Side A K B C ABC L M KLM by SSS We only had to prove that three pairs of corresponding parts are congruent!

CONGRUENT TRIANGLES by Side -Angle-Side A K B C ABC L M KLM by SAS We only had to prove that three pairs of corresponding parts are congruent!

CONGRUENT TRIANGLES by Angle-Side-Angle A B K C ABC L M KLM by ASA We only had to prove that three pairs of corresponding parts are congruent!

CONGRUENT TRIANGLES by Angle-Side A B K C ABC L M KLM by AAS We only had to prove that three pairs of corresponding parts are congruent!

SUMMARY: CONGRUENCE THEOREMS IN TRIANGLES T T R L S J K RST JKL by SSS R J RST T R J RST L L S K JKL by SAS T S K JKL by ASA R J RST L S K JKL by AAS

List the parts that are missing to be marked as congruent for both triangles to be congruent by AAS: K J R L T S If RST JKL by AAS SRT KJL STR KLJ RS KJ Missing part

List the parts that are missing to be marked as congruent for both triangles to be congruent by CPCTC: A L K B IF M C KLM by CPCTC THEN ABC AB KL AC BC KM LM ACB ABC KML KLM BAC 42 LKM Missing parts

List the parts that are missing to be marked as congruent for both triangles to be congruent by SAS: K J R L T S SR If RST JKL by SAS KJ KJL SRT RT JL Missing part

List the parts that are missing to be marked as congruent for both triangles to be congruent by SSS: K R J T S L If RST JKL by SSS SR RT KJ JL TS LK Missing part

List the parts that are missing to be marked as congruent for both triangles to be congruent by ASA: K J R L T S SRT If RST JKL by ASA RT STR KJL JL KLJ Missing part

RIGHT TRIANGLES CONGRUENT by LEG - LEG R T S J L K RST JKL by LL We only had to prove that two pairs of corresponding parts are congruent!

RIGHT TRIANGLES CONGRUENT by LEG-ANGLE R S T J K We only had to prove that two pairs of corresponding parts are congruent! RST L JKL by L A

RIGHT TRIANGLES CONGRUENT BY HYPOTENUSE - LEG R S We only had to prove that two pairs of corresponding parts are congruent! T J RST K L JKL by H L

RIGHT TRIANGLES CONGRUENT by HYPOTENUSE - ANGLE R S T J K We only had to prove that two pairs of corresponding parts are congruent! RST L JKL by HA

SUMMARY: CONGRUENCE THEOREMS IN RIGHT TRIANGLES T R T S L J RST S L K K J RST JKL by LL T R L R JKL by LA T S R L K JKL by HL J S RST K JKL by HA

List the parts that are missing to be marked as congruent for both triangles to be congruent by LA R If JKL by LA T S J RS STR K RST L KJ Missing part KLJ

HOMEWORK p. 366 # 4, 12 and 14