Geometry 1 Unit 4Congruent Triangles Casa Grande Union High School Fall 2008
Geometry 1Unit 4 4.1 Triangles and Angles
Equilateral- all 3 sides are congruent Isosceles- at least 2 sides are congruent Scalene- No sides congruent Classifying Triangles by Sides
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.
A C B Parts of triangles • Vertex (plural vertices) • The points joining the sides of a triangle • Adjacent sides • Sides sharing a common vertex • Side AB is adjacent to side BC
Exterior angle A Interior angle C B • Interior angle • Angle on the inside of a triangle • Exterior angle • Angle outside the triangle that is formed by extending one side
Triangle Sum Theorem • The sum of the three interior angles of a triangle is 180º
B 1 A C • 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
Example 2 Find the measure of each angle. 2x + 10 x x + 2
B A C D Example 3 Given that ∠ A is 50º and ∠B is 34º, what is the measure of ∠BCD? What is the measure of ∠ACB?
Hypotenuse Legs Right triangle vocabulary • Legs • Sides that form the right angle • Hypotenuse • Side opposite the right angle
Corollary to the Triangle Sum Theorem • The acute angles of a right triangle are complementary. m∠A+ m∠B = 90°
13 12 5 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?
Legs • 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.
Vertex Angle Legs Base Angles Isosceles triangle vocabulary Base
A 15 75º B C 7 Example 5 A. Given the following isosceles triangle, what is the measurement of segment AC? B. What is the measurement of angle A?
80° 53° Example 6 Find the missing measures
Example 7 Given: ∆ABC with mC = 90° Prove: mA + mB = 90°
Geometry 1Unit 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.
B Q A P C R 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.
Example 1 S E V D F T Complete the congruence statement for the two given triangles: DEF What side corresponds with DE? What angle corresponds with E?
L K A B 9 cm 91° (4x – 3) cm (5y – 12)° 86° 113° H D 6 cm C J Example 2 In the diagram, ABCD ≅ KJHL • Find the value of x. • Find the value of y.
Third Angles Theorem • If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the third angles are also congruent.
(6y – 4)° Q R A 85° (10x + 5)° P 50° C B Example 3 Given ABC PQR, find the values of x and y.
H E 58° G 58° F J Example 4 • Decide whether the triangles are congruent. Justify your answer.
M N O P Q Example 5 Given: MN ≅ QP, MN || PQ, O is the midpoint of MQ and PN. Prove: ∆MNO ≅ ∆QPO
Geometry 1Unit 4 4.3 Proving Triangles are Congruent: SSS and SAS
B A I T R G Warm-Up Complete the following statement BIG
Definitions • Included Angle • An angle that is between two given sides. • Included Side • A side that is between two given angles.
L J K P Example 1 • Use the diagram. Name the included angle between the pair of given sides.
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.
U S T V STW W Example 2 • Complete the congruence statement. • Name the congruence shortcut used.
H L M I N J HIJ LMN Example 3 • Determine if the following are congruent. • Name the congruence shortcut used.
A B C O R X XBO Example 4 • Complete the congruence statement. • Name the congruence shortcut used.
P T S Q Example 5 • Complete the congruence statement. • Name the congruence shortcut used. SPQ
Constructing Congruent Triangles C A B • 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.
M A B P Example 6 Given: AB ≅ PB, MB ⊥ AP Prove: ∆MBA ≅ ∆MBP
Example 7 Use SSS to show that ∆NPM ≅∆DFE N(-5, 1) P(-1, 6) M(-1, 1) D(6, 1) F(2, 6) E(2, 1)
Geometry 1Unit 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.
Q U QUA D A Example 1 • Complete the congruence statement. • Name the congruence shortcut used.
R M N P Q RMQ Example 2 • Complete the congruence statement. • Name the congruence shortcut used.
F B E A C D ABC FED Example 3 • Determine if the following are congruent. • Name the congruence shortcut used.
C B F D M Example 4 Given: B ≅C, D ≅F; M is the midpoint of DF. Prove: ∆BDM ≅∆CFM
Geometry 1Unit 4 4.5 Using Congruent Triangles