Chapter 4 Triangles and Congruence 4.1 Apply Triangle Sum Properties 4.2 Apply Congruence and Triangles 4.3 Prove Triangles Congruent by SSS 4.4 Prove Triangles Congruent by SAS and HL 4.5 Prove Triangles Congruent by ASA and AAS 4.6 Use Congruent Triangles 4.7 Isosceles and Equilateral Triangles SOL G.6 The student, given the information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. SOL G.7 The student, given the information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Name Block
4.1 Apply Triangle Sum Properties Classifying triangles by their ANGLES Name Words Picture Acute acute angles. Obtuse obtuse angle. Right right angle. Equiangular angles congruent. SIDES Name Words Picture Equilateral sides congruent. Isosceles congruent sides. Scalene congruent sides.
Classify a triangle in a coordinate plane. Classify RST by its sides. Then determine if the triangle is a right triangle. m A + m B + m C = The sum of the measures of the interior angles of a triangle is. m 1 = m + m The measure of an exterior angle of a triangle is equal to the sum of the measures of the two angles. Finding an angle measure. Use the diagram at the right to find the measure of DCB. m A + m B = The acute angles of a right triangle are.
Finding angle measures from a verbal description. The front face of the wheelchair ramp shown forms a right triangle. The measure of one acute angle in the triangle is eight times the measure of the other. Find the measure of each acute angle. You try! 1. Triangle JKL has vertices J( 2, 1), K(l, 3), and L(5, 0). Classify it by its sides. Then determine if it is a right triangle. 2. Find the measure of l in the diagram shown. 3. Find the value of x. 4. Find the value of x. 88 63 2x 122 x 56
5. Use the diagram to find the measures of the following angles. m 1 = m 7 = m 2 = m 8 = m 3 = m 9 = m 4 = m 10 = m 5 = m 11 = m 6 = m 12 = 6. The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building?
4.2 Apply Congruence and Triangles Same size and shape Different size or shape ALWAYS list the corresponding vertices in the same order. Congruence statement: Corresponding angles: Corresponding sides: Two polygons are congruent if and only if their corresponding parts are congruent. Example: Write a congruence statement for the following triangles.
Example: In the diagram, DEFG SPQR. a. Find the value of x. E Q R D F b. Find the value of y. P S G If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. If and, then. Example: Find m BDC. Properties of Congruent Triangles For any triangle ABC, If then If and then
Try it! Write a congruence statement for any figures that can be proved congruent. Explain your reasoning. 1. 2. 3. Find the value of x. 4. 5. In problems 6 and 7, use the given information to find the indicated values. 6. Given ABC DEF, find the values of x and y. 7. Given HJK TRS, find the values of a and b.
4.3 Prove Triangles Congruent by SSS If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. B E A C D F If Side and Side and Side, then HINT: Always look for shared sides!! Decide whether the congruence statement is true. Explain your reasoning. 1. ABD CDB 2. RST RQT 3. ABC DEF Use the given coordinates to determine if ABC DEF. 4. A(1, 2) B(4, -3) C(2, 5) D(4, 7) E(7, 2) F(5, 10) 5. A(1, 1) B(4, 0) C(7, 5) D(4, -5) E(6, -6) F(9, -1)
Determine whether ABC DEF. Explain your reasoning. 6. 7. 8. Given: AB CB, D is the midpoint of AC Prove: ABD CBD 1. 2. 3. 4. 5. Statements Reasons You try! 1. Is it possible to prove that the two triangles below are congruent using the SSS postulate? Explain your reasoning in words or by formulating a geometric proof. Given: AF CD, AB EF, BC ED
4.4 Prove Triangles Congruent by SAS, HL 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. If Side and Angle and Side, then HINT: Always look for vertical angles!! 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. If Hypotenuse and Leg, then HINT: All right angles are congruent!! Example: How could we prove these triangles congruent?
Use the diagram to name the included angle between the given pair of sides. 1. AB and BC 2. BD and DA 3. CD and DB Decide whether enough information is given to prove that the triangles are congruent. If there is enough information, state the congruence postulate or theorem you would use. 4. 5. 6. A 7. Given: B is the midpoint of AE B is the midpoint of CD Prove: ABD EBC D B C E Statements Reasons 1. 1. Given 2. 2. Definition of midpoint 3. 3. Given 4. 4. Definition of Midpoint 5. 5. Vertical Angles Theorem 6. 6. SAS Congruence Postulate
4.5 Prove Triangles Congruent by ASA AAS 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. If Angle and Side and Angle, then HINT: If lines are parallel, look for alternate interior angles!! 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. If Angle and Angle and Side, then HINT: Look for shared angles! Explain how you can prove that the indicated triangles are congruent using the given theorem or postulate. 1. BEF BED by SAS 2. ADB CFB by ASA 3. AFB CDB by AAS
4.6 Use Congruent Triangles By definition, congruent triangles have congruent corresponding parts. So, if you can prove that two triangles are congruent, you know that their corresponding parts must be congruent as well. You MUST prove triangles are congruent (using SSS, SAS, HL, ASA, or AAS) before you can use CPCTC! Example 1 To ensure that sailboat races are fair, the boats and their sails are required to be the same size and shape. A. Write a congruence statement relating the triangles in the photo. B. Name six pairs of congruent segments. C. Name six pairs of congruent angles. Example 2 In an umbrella frame, the stretchers are congruent and they open to angles of equal measures. Prove that the angles formed by the shaft and the ribs are congruent (in other words, prove 3 4). Statements Reasons
Example 3 According to legend, one of Napoleon s officers used congruent triangles to estimate the width of the river. On the riverbank, the officer stood up straight and lowered the visor of his cap until the farthest thing he could see was the edge of the opposite bank. He then turned and noted the spot on his side of the river that s was in line with his eye and the tip of his visor. Given: DEF and DEG are right angles; EDG EDF The officer then paced off the distance to his spot and declared that distance be the width of the river! Use congruent triangles to prove he is correct! Statements Reasons 1. 2. 3. 4. 5. 6. EF EG 1. 2. 3. 4. 5. 6. You try! The object of the two shown is to make the two spheres meet and strike each other repeatedly on one side of the wand and then again on the other side. If JKL MLK and JLK MKL, prove that JK ML. Statements Reasons
4.7 Use Isosceles and Equilateral Triangles Isosceles Triangle Vocabulary: Legs, Vertex Angle, Base, Base Angles them are congruent. If two sides of a triangle are congruent, then the angles opposite B If then A C opposite them are congruent. If two angles of a triangle are congruent, then the sides B If then A C If a triangle is equilateral, then it is equiangular. If there are then there are. equilateral. If a triangle is equiangular, then it is If there are then there are. Find the values of x and y. 1. 2. x 17 9y
Find the values of x and y continued... (3x + 8) 3. 4. 98 (5x + 6) (y 9) 53 (8y + 10) You try! Use the properties of isosceles and equilateral triangles to find the measure of the indicated angle. 1. 2. 3. 4. The length of YX is 20 feet. Explain why the length of YZ is the same.
Practice with SSS, SA, HL, AAS, and ASA Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, write not possible.
Sample SSS Proofs P R S 1 Q T 2 J K M L N C 3 A B D
Sample SAS Proofs 1 J M N L A 2 D C B 3 G H J I
Sample HL Proofs 1 2
1 Sample ASA Proofs 2 3
Sample AAS Proofs 1 2 3
Triangle Congruence Practice Compare the triangles and determine whether they can be proven congruent, if possible, by SSS, SAS, ASA, AAS, or HL. Write your answer the box. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.