Geometry 1 Unit 4 Congruent Triangles Casa Grande Union High School Fall 2008

Geometry 1 Unit 4 4.1 Triangles and Angles

Classifying Triangles by Sides Equilateral- all 3 sides are congruent Isosceles- at least 2 sides are congruent Scalene- No sides congruent

Classifying Triangles by Angles Acute Triangle- All angles are less than 90° Obtuse Triangle- 1 angle greater than 90 ° Right Triangle- 1 angle measuring 90°

Classifying Triangles Example 1 Name each triangle by its sides and angles A. B. C.

Parts of triangles Vertex (plural vertices) Adjacent sides The points joining the sides of a triangle Adjacent sides Sides sharing a common vertex Side AB is adjacent to side BC A B C

Interior angle Exterior angle Angle on the inside of a triangle Angle outside the triangle that is formed by extending one side A B C Interior angle Exterior angle

Triangle Sum Theorem The sum of the three interior angles of a triangle is 180º

Exterior Angle Theorem The measure of the exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles. Example: m∠1=m∠A+ m∠B B A C 1

Example 2 Find the measure of each angle. 2x + 10 x x + 2

Example 3 Given that ∠ A is 50º and ∠B is 34º, what is the measure of ∠BCD? What is the measure of ∠ACB? D A B C

Right triangle vocabulary Legs Sides that form the right angle Hypotenuse Side opposite the right angle Legs Hypotenuse

Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary. m∠A+ m∠B = 90°

Example 4 A. Given the following triangle, what is the length of the hypotenuse? B. What are the length of the legs? C. If one of the acute angle measures is 32°, what is the other acute angle’s measurement? 13 12 5

Legs Base Base Angles Vertex Angle The two congruent sides of an isosceles triangle. Base The noncongruent side of an isosceles triangle. Base Angles The two angles that contain the base of an isosceles triangle. Vertex Angle The noncongruent angle in an isosceles triangle.

Isosceles triangle vocabulary Legs Base Angles Vertex Angle Base

Example 5 A B C 75º 15 7 A. Given the following isosceles triangle, what is the measurement of segment AC? B. What is the measurement of angle A?

Example 6 Find the missing measures 80° 53°

Example 7 Given: ∆ABC with mC = 90° Prove: mA + mB = 90° Statement Reason 1. mC = 90° 2. mA + mB + mC = 180° 3. mA + mB + 90° = 180° 4. mA + mB = 90°

4.2 Congruence and Triangles Geometry 1 Unit 4 4.2 Congruence and Triangles

Congruent Figures Congruent Figures Figures are congruent if corresponding sides and angles have equal measures. Corresponding Angles of Congruent Figures When two figures are congruent, the angles that are in corresponding positions are congruent. Corresponding Sides of Congruent Figures When two figures are congruent, the sides that are in corresponding positions are congruent.

Congruent Figures For the triangles below, ∆ABC ≅ ∆PQR The notation shows congruence and correspondence. When writing congruence statements, be sure to list corresponding angles in the same order. A B C Corresponding Angles Corresponding Sides A ≅ P AB ≅ PQ B ≅ Q BC ≅ QR C ≅ R CA ≅ RP P Q R

Example 1 Complete the congruence statement for the two given triangles: DEF What side corresponds with DE? What angle corresponds with E? S V T D E F

Example 2 In the diagram, ABCD ≅ KJHL Find the value of x. Find the value of y. J H K L (5y – 12)° (4x – 3) cm A B D C 9 cm 6 cm 86° 91° 113°

Third Angles Theorem If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the third angles are also congruent.

Example 3 (6y – 4)° Q R A 85° (10x + 5)° P 50° C B Given ABC  PQR, find the values of x and y. A B C P Q R 85° 50° (6y – 4)° (10x + 5)°

Example 4 Decide whether the triangles are congruent. Justify your answer. F H E G J 58°

Example 5 Given: MN ≅ QP, MN || PQ, O is the midpoint of MQ and PN. Prove: ∆MNO ≅ ∆QPO Statements Reasons 1. 2. Alt. Interior Angles Theorem 3. Vertical Angles Theorem 4.O is the midpoint of MQ and PN 5. Def of Midpoint 6. ∆MNO ≅ ∆QPO P M O Q N

4.3 Proving Triangles are Congruent: SSS and SAS Geometry 1 Unit 4 4.3 Proving Triangles are Congruent: SSS and SAS

Warm-Up Complete the following statement BIG  B I G R A T

Definitions Included Angle Included Side An angle that is between two given sides. Included Side A side that is between two given angles.

Example 1 Use the diagram. Name the included angle between the pair of given sides. K P J L

Triangle Congruence Shortcut SSS If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

Triangle Congruence Shortcuts SAS If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Example 2 U S T V STW   W Complete the congruence statement. Name the congruence shortcut used. S T U V W STW  

Example 3 H L M I N HIJ  LMN J Determine if the following are congruent. Name the congruence shortcut used. J H I L M N HIJ  LMN No. Triangle HIJ is congruent to MNL

Example 4 A B C O R X XBO   Complete the congruence statement. Name the congruence shortcut used. B O X C A R XBO  

Example 5 P T S Q Complete the congruence statement. Name the congruence shortcut used. SPQ   S P Q T

Constructing Congruent Triangles Construct segment DE as a segment congruent to AB Open your compass to the length of AC. Place the point of your compass on point D and strike an arc. Open the compass to the width of BC. Place the point of your compass on E and strike an arc. Label the point where the arcs intersect as F. A C B

Example 6 Given: AB ≅ PB, MB ⊥ AP Prove: ∆MBA ≅ ∆MBP Statements Reasons 1. MB ⊥ AP 2. Perpendicular lines form right angles 3. Right angles are congruent 4. AB ≅ PB 5. MB ≅ MB 6.

Example 7 Use SSS to show that ∆NPM ≅∆DFE N(-5, 1) P(-1, 6) M(-1, 1)

4.4 Proving Triangles are Congruent: ASA and AAS Geometry 1 Unit 4 4.4 Proving Triangles are Congruent: ASA and AAS

Triangle Congruence Shortcuts ASA If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Triangle Congruence Shortcuts AAS If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.

Example 1 Q U A D QUA   Complete the congruence statement. Name the congruence shortcut used. Q U A D QUA  

Example 2 Complete the congruence statement. Name the congruence shortcut used. RMQ   M R Q N P

Example 3 A B C F E D ABC  FED Determine if the following are congruent. Name the congruence shortcut used. A B C F E D ABC  FED

Example 4 Given: B ≅C, D ≅F; M is the midpoint of DF. Prove: ∆BDM ≅∆CFM Statements Reasons 1. 2. 3. Def of Midpoint 4.

4.5 Using Congruent Triangles Geometry 1 Unit 4 4.5 Using Congruent Triangles

Warm-up State which postulate or theorem you can use to prove that the triangles are congruent. Then, write the congruence statement. C G H S

Example 1 Given: NO is parallel to MP, MN is parallel to PO Prove MN = OP (Prove Δ MNO Δ OPM) Mark the given information first Then, mark the deduced information Statements Reasons 1.NO||MP, MN|| PO 1. Given 2. 3. 4. 5. 6.

Example 2 H J L K Given: HJ || KL, JK || HL Prove: LHJ ≅ JKL

Example 3 Given: MS || TR, MS ≅ TR Prove: A is the midpoint of MT. Statements Reasons 1. 2. 3. 4. 5. 6.

4.6 Isosceles, Equilateral, and Right Triangles Geometry 1 Unit 4 4.6 Isosceles, Equilateral, and Right Triangles

Warm-Up 1 Find the measure of each angle. 90° 30° 60° a b

Warm-Up 2 Find the measure of each angle. 110 150 90

Isosceles triangle theorems Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are also congruent. Converse of the Base Angles theorem If two angles of a triangle are congruent, then the sides opposite them are also congruent

Example 1 35° x

Example 2 15° b a

Example 3 Find each missing measure 63° 10 cm m n p

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

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

Example 4 Find the value of x 12 in 2x in

Example 5 Find the value of x and y. y x

Example 6 Find the value of x and y. 75° x° y°

4.7 Triangles and Coordinate Proof Geometry 1 Unit 4 4.7 Triangles and Coordinate Proof

Warm-up What is the midpoint formula? What is the distance formula? What are some postulates and theorems you have learned about triangles this chapter?

Vocabulary Coordinate Proof A proof involving placing geometric figures on a coordinate plane. Uses the midpoint formula, distance formula, postulates and theorems to prove statements about the figure

Placing Figures in a Coordinate Plane Complete the activity on p. 243 individually. Compare your results to those of your partners. What did you learn?

Example 1 A right triangle has legs of 3 units and 4 units. Place the triangle on a coordinate grid. Label the vertices, then find the length of the hypotenuse. 3 4

Example 2 In the diagram, ΔABO ≅ ΔCBO. Find the coordinates of point B. C(10,0) A(0,10) O(0,0) B

Example 3 Write a plan to prove that OU bisects TOV. U(0,5) T(-3,5)

Example 4 Find the coordinates of P. P N(h,0) M(0,k)

Constructions review Duplicate the given triangle. Write the steps that you used to construct the new triangle