Congruent Triangles Unit 3

Bellringer Find the equation of the line that has a slope of -5 and a y-intercept of 7. Find the equation of the line that has a slope of 2/3 and goes through the point (-9,-3).

Apply Triangle Sum Properties 4.1 I CAN classify triangles and find measures of their angles.

Triangle – a polygon with three sides

Example 1: Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles.

Ways to Measure Side Lengths and Angles Ruler - Sides Protractor – Angles Distance Formula – Sides Right Angles - Slopes

Example 2:Classifying Triangles Classify ABC by its sides and by its angles.

Vocabulary Interior Angles: Angles inside the polygon Exterior Angles: The angles that form linear pairs with the interior angles

Theorem 4.1 – Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180⁰.

Example 4: Find the measure of each interior angle of ABC, where m∠A = x⁰, m∠B = 2x⁰, and m∠C = 3x⁰. Example 5: Find the measures of the acute angles of the right triangle in the diagram shown.

Corollary to a theorem – a statement that can be proved easily using the theorem Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary. m∠A + m∠B = 90⁰ A C B

Theorem 4.2 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

Example 6 Find m∠DEF.

Example 7: Find the measure of 1 in the diagram shown.

Exit Slip

Class Work/Homework Page 221 #1-6,8-10,14-16

Congruence and Triangles 4.2 I CAN define congruent figures that have the same shape and size.

Congruent Figures – all parts of one figure are congruent to the corresponding parts of the other figure

Congruence Statements Always list the corresponding vertices in the same order Can be written in more than one way

Example 1 Write a congruence statement for the triangles shown. Identify all pairs of congruent corresponding parts.

Example 2 In the diagram, ABCD ≅ FGHK. Find the value of x. Find the value of y.

Example 3 If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape? Explain.

Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

Example 4 Find m∠YXW.

Theorems B C E F K L

Exit Slip

Class Work/Homework Page 228 #3-17; 19-21, 23, 26, 28

Bell Fun: Warm-Up 4.3 Write a congruence statement. How do you know that ∠N ≅∠R? Find x.

Triangles Congruence by SSS 4.3 I CAN demonstrate that when corresponding sides are congruent the triangles must be congruent. I CAN prove theorem about triangles.

Postulate Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

Examples Decide whether the congruence statement is true. Explain your reasoning.

Example 4. Use the given information to determine if JKL ≅ RST. J(-3, -2), K(0, -2), and L(-3, -8) R(10, 0), S(10, -3), and T(4, 0) y x

Class Work/Homework Page 236 #1-10, 13-19, 24, 26

Bell Fun: Warm – Up 4.4

Triangle Congruence by SAS & HL 4.4 I CAN demonstrate that when corresponding sides are congruent the triangles must be congruent. I CAN prove theorems about triangles. I CAN apply theorems, postulates, or definitions to prove theorems about triangles.

Included angle – two sides forming an angle

Postulate Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

Example 1

Right Triangle Legs – the sides adjacent to the right angle; the sides forming the right angle Hypotenuse – the side opposite of the right angle

Theorem Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

Example 2

Examples Name the congruent triangles. Explain. 1. 2. 3.

Closing Question: What do you need to prove that triangles are congruent?

Exit Slip

Class Work/Homework Page 243 #1-14, 16-17, 20-22, 25-27, 32 – 38 even

Bell Fun: Warm-Up 4.5 Tell whether the pair of triangles are congruent or not and why.

Triangle Congruence by ASA & AAS 4.5 I CAN demonstrate that when corresponding sides and angles are congruent the triangle must be congruent. I CAN list the sufficient condition to prove triangles are congruent. I CAN prove theorems about triangles. I CAN apply theorems, postulates, or definitions to prove theorems about triangles.

Included side – the side connecting vertices of two angles

Postulates Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

Theorem Angle-Angle-Side (AAS) Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

Example 1 Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. b.

CPCTC Corresponding parts of congruent triangles are congruent

Exit Slip

Class Work/Homework Page 252 #3-20, 32, 34

Bell Fun: Warm-Up 4.7 Classify each triangle by its sides. 2 cm, 2 cm, 2 cm 7 ft, 11 ft, 7 ft 9 m, 8 m, 10 m In ABC, if m∠A = 70⁰ and m∠B = 50⁰, what is m∠C? In DEF, if m∠D = m∠E and m∠F = 26⁰, what are the measures of ∠D and ∠E?

Isosceles & Equilateral Triangles 4.7 I CAN use theorems about isosceles and equilateral triangles.

Vertex Angle Isosceles Triangles leg leg Base angles base Isosceles Triangle – a triangle where at least two sides are congruent Legs of An Isosceles Triangle – two congruent sides Vertex Angle – the angle formed by the legs Base of Isosceles Triangle – the third side of the triangle; the noncongruent side Base Angles – two angles adjacent to the base

Theorems

Examples Copy and complete the statement.

Examples 3. In PQR, PQ ≅ QR. Name two congruent angles.

Examples 4. Find the measures of ∠X and ∠Z.

Corollaries

Examples 5. Find ST in the triangle at the right. 6. Is it possible for an equilateral triangle to have an angle measure other than 60⁰? Explain.

Examples 7. Find the values of x and y in the diagram.

Exit Slip

Class Work/Homework Page 267 #3-20, 24, 38,