G E O M E T R Y

CHAPTER 4

CONGRUENT TRIANGLES

Notes & Study Guide

2 TABLE OF CONTENTS

TRIANGLES AND ANGLES ................................................................................ 3 CONGRUENCE AND TRIANGLES ..................................................................... 7 PROVING TRIANGLES CONGRUENT (SSS/SAS) ............................................ 9 PROVING TRIANGLES CONGRUENT (ASA/AAS)............................................ 9 ISOSCELES, EQUILATERAL & RIGHT TRIANGLES ...................................... 12 EXTRA HOMEWORK EXAMPLES ................................................................... 14 < end of page >

TRIANGLES AND ANGLES 3

INTRODUCTION In Chapter 4 we work with our first triangle chapter. We will first establish the types of triangles and their vocabulary. Then we will study the concept of congruence and how it is applied to triangles.

When we classify (categorize) triangles we do it by looking at the triangles’ sides and/or its’ angles. CLASSIFYING TRIANGLES BY SIDES To classify a triangle by its sides, find out how many of the sides are congruent. Equilateral all 3 sides are congruent Isosceles at least 2 sides are congruent Scalene no sides are congruent CLASSIFYING TRIANGLES BY ANGLES To classify a triangle by its angles, check the sizes of the angles and whether they are acute, right or obtuse. Acute triangle all 3 angles are acute (< 90°) Right triangle must have 1 right angle (= 90°) Obtuse triangle must have 1 obtuse angle (> 90°) Equiangular triangle type of acute triangle where all 3 angles are equal < end of page >

All triangles get

classified both ways at the same time!

4 TRIANGLES AND ANGLES

MORE IMPORTANT TERMS The following terms are important when discussing triangles and we will use these throughout the rest of the course… Vertex Each of the three “corners” of the triangle Side Each of the three segments of the triangle Adjacent Sides any two sides of a triangle that share a vertex (they touch that vertex) Opposite Side the side that is directly across from a particular angle in a triangle (does not touch that vertex)

Shown here: CA and BA are adjacent to angle A while CB is opposite of A. RIGHT TRIANGLES Hypotenuse – the longest side of a right triangle (it is ALWAYS across from the right angle!) Legs – the sides that form the right angle in a right triangle ISOSCELES TRIANGLES Legs – the two congruent sides of an isosceles triangle Base – the third (non-congruent) side of an isosceles triangle < end of page>

TRIANGLES AND ANGLES 5

MAIN TRIANGLE THEOREMS As we study triangles, we will make use of a couple of key principles. These theorems will be used repeatedly, so it’s important to know them well. But first… Interior angles – angles on the inside Exterior angles – angles on the outside

THEOREM 4.1 (Triangle Sum Theorem) The sum of the measures of the interior angles of a triangle is ALWAYS 180°. Shown: angle A + angle B + angle C = 180

THEOREM 4.2 (Exterior Angle Theorem) The measure of an exterior angle of a triangle is equal to the sum of the two interior angles across from it. Shown: angle 1 = angle A + angle B < end of page >

6 TRIANGLES AND ANGLES

HOMEWORK EXAMPLES Classify each triangle. ••• Sides of 2cm, 3cm, 4cm ••• Angles of 20°, 145°, 15° Find the measures of the numbered angles. Find the measure of the third angle of a triangle given the first two. ••• Angle 1 = 56° Angle 2 = 42° Angle 3 = ____ ••• Angle 1 = 113° Angle 2 = 44° Angle 3 = ____ Find the value of x in each figure. < end of page >

CONGRUENCE AND TRIANGLES 7

CONGRUENCE When two figures have the same size AND the same shape, they are called congruent. To be the same size…all the sides have to match up To be the same shape…all the angles have to match up Each pair of sides or angles that match up to each other are called corresponding sides or angles.

Example: the tick marks show which parts match up (correspond) to each other… Sides: AB = PQ, BC = QR, CA = RP Angles: A = P, B = Q, C = R

In order for two triangles to be congruent, a total of 6 things must happen… all 3 pairs of sides must match AND all 3 pairs of angles must match Look for tick marks, angle marks and labeled angles to help you match things up. NAMING CONGRUENT FIGURES When you name figures that are congruent, the order of the letters MUST match up (correspond) to each other. (even when you read the names, too) Example: Let’s say the first triangle is DABC .

Since A @ P , B @Q & C @ R , the name of the

other triangle MUST be named as DPQR .

So, if the 1st one was DBCA , then the second one must be ______.

And, if the 1st one was DCAB , then the second one must be ______.

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8 CONGRUENCE AND TRIANGLES

WRITING A CONGRUENCE STATEMENT A congruence statement is a sentence that just says that one figure is congruent to another (by name). Writing one is very easy… (1) Name the 1st figure (usually many ways to do it) (2) Name the 2nd figure so that the letters correspond (match

up) to the 1st one. (3) Put the congruent symbol in between the names. Example: One possible congruence statement for this is Rearrange the name of the first triangle and list some more possible answers. HOMEWORK EXAMPLES Shown at right: DABC @ DTVU

• ÐA @ ____ • VT @ ____

• BC = _____ • DUTV @ ____

Decide if the two figures are congruent. If they are, write a congruence statement for them. Assume the figures shown are congruent. Find the values of the variables. < end of page >

DDEF @ DJKL

PROVING TRIANGLES CONGRUENT 9

INTRODUCTION In section 2, we learned that for two triangles to be congruent we had to have 6 things happen: 3 pairs of congruent sides and 3 pairs of congruent angles. We’d rather not have to do 6 things to show that two triangles are congruent. So in sections 3 & 4, we will learn about 4 shortcut Postulates that let us do it in fewer steps.

TRIANGLE CONGRUENCE POSTUALTES To prove these are congruent… …we need to have all of this.

To make it easier, we use Triangle Congruence Postulates to allow us to prove the same congruence, but with fewer requirements (usually just 3).

POSTULATE 19 •• Side-Side-Side (SSS) Congruence Postulate If 3 sides of one triangle are congruent to 3 sides of another triangle, then the two triangles are congruent. You don’t need the angles at all. If you’ve got the 3 pairs of sides, you got it.

POSTULATE 20 •• Side-Angle-Side (SAS) Congruence Postulate If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent. Here you need any 2 sides and the angle that is included (in between) the two sides.

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10 PROVING TRIANGLES ARE CONGRUENT

POSTULATE 21 •• Angle-Side-Angle (ASA) Congruence Postulate

If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent. Here you need any 2 angles and the side that is included (in between) the two angles.

THEOREM 4.5 •• Angle-Angle-Side (AAS) Congruence Postulate If 2 angles and the non-included side of one triangle are congruent to 2 angles and the non-included side of another triangle, then the triangles are congruent. You need any 2 angles and the side that comes right after one of the angles.

THEOREM 4.8 •• Hypotenuse-Leg (HL) Congruence Postulate If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and same leg of another right triangle, then the triangles are congruent. Right triangles only! You need the hypotenuse and the same leg.

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PROVING TRIANGLES ARE CONGRUENT 11

HOMEWORK EXAMPLES Decide whether enough information is given to prove that the triangles are congruent. If yes, identify which postulate or theorem you would use. < end of page >

12 ISOSCELES, EQUILATERAL & RIGHT TRIANGLES

INTRODUCTION In this section, we will look at the most common features of isosceles, equilateral and right triangles. Knowing these features will help make all work with triangles easier because these features are almost exclusively used as shortcuts.

ISOSCELES TRIANGLES In an isosceles triangle, two sides (the legs) are always congruent. Not only are the two legs congruent, the two base angles (shown in red) are also congruent. Notice that base angles are directly across (opposite) from the congruent legs. This pattern is consistent for all triangles and makes a theorem.

THEOREM 4.6 •• Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are also congruent.

** This theorem is reversible (Converse Theorem 4.7)

EQUILATERAL TRIANGLES Equilateral triangles have all sides congruent and the Base Angle Theorem works on them too. Since all 3 sides are equal, then the angles opposite those sides are equal too. When all the angles of any shape are congruent, we call it equiangular.

If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral.

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ISOSCELES, EQUILATERAL & RIGHT TRIANGLES 13

HOMEWORK EXAMPLES Find the values of the variable(s) in the triangles. < end of page >

14 EXTRA HOMEWORK EXAMPLES

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