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.

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

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

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. Key Concepts

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.

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.

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

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.

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

Using Geogebra to solve this problem:

p. 361# 4-20 (mo4)

C. N. Colόn Geometry Reference: SIMON PEREZ.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

p. 366 # 4, 12 and 14