Chapter Four. Congruent Triangles
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- Candace Pitts
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1 Chapter Four Congruent Triangles
2 Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates and theorems in this chapter. D. Identify the corresponding parts of congruent figures. E. Use pairs of congruent triangles to write proofs about angles and sides. F. Use relationships between angles and sides of triangles to prove congruence of triangles.
3 Section 4-1 Congruent Figures Homework Pages : 1-21
4 Objectives A. Use the term congruent correctly when referring to figures. B. Recognize the conditions that make two figures congruent. C. Rotate figures about several axes to determine congruency. D. Recognize corresponding sides and corresponding angles.
5 The Confluence of Congruence Congruent Figures with the same SIZE and SHAPE! Congruent line segments two or more line segments with the same length. Congruent angles two or more angles with the same measure.
6 Corresponding Parts When we talk about corresponding parts of polygons, we are basically matching each part of one polygon to exactly one part of another polygon. We can speak of: Corresponding vertices Corresponding sides Corresponding angles Corresponding parts do NOT necessarily have to be in the same relative location (upper right, lower left, etc.) However, it may be helpful to ROTATE the figures to place corresponding parts in the same relative locations.
7 Checking Congruence of Polygons Congruent polygons polygons are congruent if and only if the vertices can be matched up so the corresponding parts (sides and angles) are congruent. Congruent triangles triangles are congruent if and only if the vertices can be matched up so that corresponding parts (sides and angles) are congruent.
8 Methods for Checking Congruence of Polygons One-to-one matching show that each side and angle of one polygon is congruent to each corresponding side and angle of the other polygon. NOTE: ORDER MATTERS! A quadrilateral with consecutive sides of length 4, 5, 6, and 7 is NOT congruent to a quadrilateral with consecutive sides of length 4, 6, 5, 7. Rotating a polygon Rotate a polygon clockwise or counterclockwise until each segment and each angle is congruent to the corresponding angles and segments AND in the same relative position of the second polygon. Flipping a polygon Rotate a polygon across an axis until each segment and each angle is congruent to the corresponding angles and segments AND in the same relative position of the second polygon.
9 One-to-one Matching A 1 2 B E 5 6 F 3 4 C D 7 8 G H AB EF 1 5 BD FH 2 6 DC HG 4 8 CA GE 3 7
10 A G H Rotating a Polygon B H G C D F E E E F G H F AB EF BD FH DC HG CA GE
11 Corresponding Parts of Congruent Polygons Flipping Polygons These two polygons are congruent because the left polygon can be flipped over on top of the right polygon and all of the sides and angles will match up These two polygons are not congruent because you cannot get both the sides and the angles to match up at the same time. You can get one to work but not the other, 3
12 Corresponding Parts of Congruent Triangles A D B C E F D ABC D DEF if and only if A D, B E, C F AB = DE, BC = EF, AC = DF 3
13 Sample Problems Section 4-1 Suppose D BIG D CAT. Complete. 1. G 3. BI = 5. D IGB 7. If D DEF D RST, m D = 100, and m F = 40, name four congruent angles. 9. Suppose D LXR D FNE. List six congruencies that can be justified by the following reason: Corresponding Parts of Congruent Triangles.
14 Sample Problems Section The two triangles shown are congruent. Complete. a. D PAL b. PA = c. 1 because d. 2 because Then because P 1 2 A 4 3 L R
15 Sample Problems Section Plot the given points on a piece of graph paper. Draw D FAT. Locate point C so that D FAT D CAT. F(7, 5) A(- 2, 2) T(5, 2) Plot the given points on graph paper. Draw D ABC and D DEF. Copy and complete the statement D ABC 15. A(- 7, - 3) B(- 2, - 3) C(- 2, 0) D(0, 1) E(5, 1) F(0, - 2) 17. A(1, 1) B(8, 1) C(4, 3) D(3, - 7) E(5, - 3) F(3, 0)
16 Sample Problems Section Plot the points on graph paper. Draw D ABC and DE. Find two locations of point F such that D ABC D DEF. A(- 1, 0) B(- 5, 4) C(- 6, 1) D(1, 0) E(5, 4) 21. In your own words explain why each of the following statements must be true. a. O is the midpoint of NM. b. NOR MOR c. RO NM N O M R E A
17 Section 4-2 Some Ways to Prove Triangles Congruent Homework Pages : 1-24
18 Objectives A. State postulates of congruence of triangles correctly. B. Apply postulates of congruence of triangles correctly. C. Distinguish between SSS, SAS, and ASA congruence. D. Correctly interpret and utilize include sides and included angles.
19 Postulate 12 (side-side-side of SSS) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. B E A C D F D ABC D DEF
20 Included Sides and Angles In a triangle, we say a side is included if it is between two referenced angles. In a triangle, we say an angle is included if it is between two referenced sides. A 1 Side AC is included between angles 1 and 3. Angle 2 is included between sides AB and BC. B 2 3 C
21 Postulate 13 (side-angle-side or 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. B E A C D F D ABC D DEF
22 Postulate 14 (angle-side-angle or 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. B E A C D F D ABC D DEF
23 Proof Given AB CD and BD AC, A B provedabc DBDC Given AB CD and BD AC Reflexive property of congruence. BC BC Postulate 12 If 3 sides of 1 DABC DBDC triangle are congruent to 3 sides of another triangle, then the triangles are congruent. C D
24 Sample Problems Section 4-2 Decide whether you can deduce by SSS, SAS, or ASA that another triangle is congruent to D ABC. If so, write the congruence and name the pattern used. If not, write no congruence 1. A C 3. K C B P N Y A B
25 Sample Problems Section X A 7. A P B C Q Y Z B C
26 Sample Problems Section A G C A B B C S T
27 Sample Problems Section B C 15. N A A F M C B
28 Sample Problems Section Given: RS ST; TU ST; R S V is the midpoint of ST Prove: D RSV D UTV 1. RS ST; TU ST m S = 90, m? = S T V is the midpoint of ST SV? RVS? D? D? 7. T V U
29 Sample Problems Section Given: E is the midpoint of Prove: D TEM D PER TP & MR T R 21. Given: Plane M bisects Prove: D POA D POB AB; PO AB M E P P A M O B
30 Sample Problems Section 4-2 Draw and label a diagram. List in terms of the diagram, what is given and what is to be proved. Then write the two column proof. 23. In an isosceles triangle, if a segment is drawn from the vertex of the angle between the congruent sides to the midpoint of the opposite side, then congruent triangles are formed. 25. If pentagon ABCDE is equilateral and has right angles at B and E, then diagonals AC and AD form congruent triangles.
31 Section 4-3 Using Congruent Triangles Homework Pages : 1-14
32 Objectives A. Understand and apply the concept of a line perpendicular to a plane. B. Apply SSS, SAS, and ASA relationships to prove congruence of angles or segments.
33 line perpendicular to a plane: a line is perpendicular to a plane if and only if they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection.
34 Using Congruent Triangles Ways to Prove Two Segments or Two Angles Congruent: 1. Identify two triangles in which the angles or segments would be corresponding parts. 2. Prove the triangles congruent. 3. State the angles or segments are congruent by Congruent Parts of Congruent Triangles (CPCT).
35 Proof Given AB and CD bisect each other at M, prove AD BC. 1. AB and CD bisect each 1. Given A C M D other at M. 2. M is the midpoint of 2. Definition of segment bisector AB and CD Definition of midpoint. AM MB; DM MC B
36 Proof Given AB and CD bisect each other at M, prove AD BC. A C M D Vertical angles are congruent. AMD BMC B Side-angle-side (SAS) postulate. DAMD DBMC Corresponding parts of congruent A B triangles are congruent If 2 lines are cut by transversal AD BC and alt int angle are congruent, then the lines are parallel.
37 Sample Problems Section Given: P S; O is the midpoint of Prove: O is the midpoint of RQ PS 1. P S O is the midpoint of PS PO SO POQ SOR D POQ D SOR QO RO O is the midpoint of RQ 7.
38 Sample Problems Section Given: WO ZO; XO YO Prove: W Z W Z O X Y
39 Sample Problems Section Given: SK NR SN KR Prove: SK NR; SN KR K S R N
40 Sample Problems Section Given: AD ME MD BE M is the midpoint of AB Prove: MD BE D E A M B
41 Sample Problems Section Given: Prove: PQ PS; 3 4 QR SR; Q 1 2 P R S
42 Sample Problems Section Given: WX YZ; 1 2; UX VX Which one(s) of the following statements must be true? 1 XW UV 2 UV YZ 3 VX UX U W V Y X Z
43 Sample Problems Section Given: RS plane Y; TRS Prove: D RTV is isosceles VRS R S Y T V
44 Section 4-4 The Isosceles Triangle Theorems Homework Pages : 1-30
45 Objectives A. Apply the terms vertex angle, legs, base, and base angles in conjunction with isosceles triangles. B. Understand and apply the theorems and corollaries associated with isosceles triangles. C. Apply corollaries that relate isosceles and equilateral triangles.
46 Parts of an Isosceles Triangle leg vertex angle base angles base leg legs: congruent sides of an isosceles triangle vertex angle: angle formed by the legs of an isosceles triangle base: side opposite the vertex angle of an isosceles triangle base angles: angles formed by a leg and the base of an isosceles triangle
47 Theorem 4-1 If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
48 Corollary 1 Theorem 4-1 An equilateral triangle is also equiangular.
49 Corollary 2 Theorem 4-1 An equilateral triangle has three 60 angles
50 Corollary 3 Theorem 4-1 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.
51 Theorem 4-2 If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
52 Corollary 1 Theorem 4-2 An equiangular triangle is also equilateral.
53 Find the value of x. Sample Problems Section x x x - 8 2x + 7 x
54 Sample Problems Section 4-4 Place the statements in the appropriate order for a proof. 9. Given: RS RT Prove: 3 4 (a) 3 4 R (b) 3 1; 2 4 (c) RS RT (d) 1 2 S T 4
55 Sample Problems Section Prove Theorem Given: M is the midpoint of JK; 1 2 Prove: JG MK J G 1 2 M K
56 Sample Problems Section Given: PQ PR; TR TS Which one(s) of the following must be true? 1 ST QP 2 ST QP P 3 T P Q R S T
57 Sample Problems Section Given: XY XZ; OY OZ Prove: m 1 = m 4 X O Y Z
58 Sample Problems Section Given: Prove: AB AC; AL and AM trisect AL AM BAC B L M C 1 2 A 3
59 Sample Problems Section Given: 3 4; Prove: 5 6 OP OQ P O Q S 5 6 R
60 Sample Problems Section Complete. a. If m 1 = 20 then m 3 =? m 4 =? and m 5 =? b. If m 1 = x then m 3 =?
61 Sample Problems Section a. If m 1 = 23, find m 7. b. If m 1 = k, find m
62 Sample Problems Section 4-4 Find the values of x and y. 27. In equiangular D ABC, AB = 4x - y, BC = 2x + 3y, and AC = In D JKL, JK = JL, m J = 2x - y, m K = 2x + 2y, and m L = x + 2y.
63 Section 4-5 Other Methods of Proving Triangles Congruent Homework Pages : 1-19 (no 9, 10)
64 Objectives A. Understand and apply the angleangle-side (AAS) theorem. B. Understand and apply the hypotenuse-leg (HL) theorem. C. Properly choose between the various ways to prove two triangles congruent. D. Understand and apply the terms hypotenuse and leg in reference to right triangles.
65 Right Triangle Definitions Remember a right triangle is a triangle that contains exactly one right angle. Hypotenuse side opposite the right angle of a right triangle. Leg sides forming the right angle of a right triangle. Leg Hypotenuse Leg
66 Theorem 4-3 (angle-angle-side) If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. B E A C D F D ABC D DEF
67 Theorem 4-4 (hypotenuse-leg) If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. B E A C D ABC D DEF D F
68 Ways to Prove Two Triangles Congruent You have postulates, corollaries, definitions, and theorems that allow you to CONCLUDE two triangles are congruent: SSS Side-side-side SAS Side-angle-side ASA Angle-side-angle AAS Angle-angle-side HL Hypotenuse-leg (right triangles ONLY!) CPCT Corresponding Parts of Congruent Triangles These are major theorems that will be used extensively for the rest of your mathematical lives! These are generally referred to by their name, but you need to know what they say, when to apply them, and how to apply them!
69 Sample Homework Problems Section Given: W and Y are rt. s; Prove: WZ YZ WX YX X 1. W and Y are rt. s 2. D XWZ & D XYZ are rt D s 3. WX YX 4.? 5. D XWZ D? 6.? W Z Y
70 Sample Homework Problems Section Given: EF EG; HG Prove: H F EG; EH GF E F H G
71 Sample Homework Problems Section 4-5 Name the method you would use to prove D AOB D AOC. Do not write the proof. 5. Given: AO plane M; BO CO A 7. Given: AO plane M; AB AC M O C B
72 Sample Homework Problems Section Given: PR PQ; SR TQ Prove: QS RT 9a. Write a two column proof that uses D RQS D QRT by SAS 9b. Write a two column proof that uses D PQS D PRT by SAS P S T R Q
73 Sample Homework Problems Section 4-5 Tell which pair of congruent parts and what method you would use to prove the triangles congruent. 11. Given: 1 2; 3 4; QR TS Prove: D QPR D TPS P 13. Given: 3 4; 5 6 Prove: D QPY D TPX Q 3 X Y T R 1 2 S
74 Sample Homework Problems Section Given: 1 2 3; Prove: 4 5 EN DG M 5 G E D N
75 Sample Homework Problems Section 4-5 Draw and label a diagram. List in terms of the diagram what is given and what is to be proved. Then write a two column proof. 17. If segments are drawn from the endpoints of the base of an isosceles triangle perpendicular to the opposite legs, then those segments are congruent. 19. If segments are drawn from the midpoints of the legs of an isosceles triangle perpendicular to the base, then those segments are congruent.
76 Section 4-6 Using More Than One Pair of Congruent Triangles Homework Pages : 1-12
77 Objectives A. Recognize and apply the transitive property of congruence. B. Recognize and apply the reflexive property of congruence. C. Prove 2 triangles congruent by first proving 2 other triangles congruent.
78 Using Congruent Triangles Ways to Prove Two Segments or Two angles Congruent: Identify two triangles in which the angles or segments would be corresponding parts. Prove the triangles congruent. State the angles or segments are congruent by C.P.C.T.
79 Proof U Given 1 2; 3 provetu TW. 4, T S V W givens 1 2; Reflexive property of congruence. SV SV Angle-side-angle postulate. DUSV DWSV Corresponding parts of UV WV corresponding triangles are congruent.
80 Proof U Given 1 2; 3 provetu TW. 4, T S V W Reflexive property of congruence. TV TV Side-angle-side postulate. DTUV DTWV Corresponding parts of congruent triangles are congruent. TU TW
81 Sample Problems Section 4-6 Give the reasons for the steps from the proof. 1. Prove: NE OS R a. D RNX D LOY b. X Y c. D NEX D OSY d. NE OS N L E X S O Y
82 Sample Problems Section Prove: G T a. D RAJ D NAK b. RJ NK R c. D GRJ D TNK d. G T A K T G J N
83 Sample Problems Section Prove: DX EX a. D POD D POE b. PD PE c. D PDX D PEX d. DX EX P E O X D
84 Sample Problems Section Given: LF KF; LA KA Prove: LJ KJ L F 1 2 J A K
85 Sample Problems Section Given: D RST D XYZ SK bisects RST; YL bisects XYZ Prove: SK YL R K S X T L Y Z
86 Sample Problems Section Given: HDE and KFG are rt. s DE FG; GD EF Prove: DH FK D G H K E F
87 Sample Problems Section Draw two line segments KL and MN, that bisect each other at O. Mark a point P on segment KN and let Q be the point where PO intersects ML. Prove that O is the midpoint of PQ.
88 Section 4-7 Medians, Altitudes and Perpendicular Bisectors Homework Pages : 1-20
89 Objectives A. Understand and apply the term median. B. Understand and apply the term altitude. C. Understand and apply the term perpendicular bisector. D. Properly apply the theorems relating to medians, altitudes and perpendicular bisectors.
90 altitude: segment from a vertex perpendicular to the line containing the opposite side median: segment from a vertex to the midpoint of the opposite side perpendicular bisector: line, segment or ray perpendicular to a segment at its midpoint
91 Altitudes Every triangle has three altitudes. 3
92 Medians Every triangle has three medians. 4
93 Perpendicular Bisectors You can only have perpendicular bisectors for segments. However, that segment may live in a plane or a ray. 4
94 Theorem 4-5 If a point lies on the perpendicular bisector of a segment, then that point is equidistant from the endpoints of the segment. C A B AC = BC
95 Theorem 4-6 If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. C A B AC = BC
96 Theorem 4-7 If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. A B O C
97 Theorem 4-8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. A B O C
98 Sample Problems Section 4-7 1a. Draw a large scalene triangle D ABC. Carefully draw the bisector of A, the altitude from A, and the median from A. These three should all be different. 1b. Draw a large isosceles triangle D ABC with vertex angle A. Carefully draw the bisector of A, the altitude from A, and the median from A. Are these three different? 3. Draw a right triangle. Then draw its three altitudes using a different color. 5. Draw a large scalene right triangle. Then draw the perpendicular bisectors of its three sides and tell whether they appear to meet in a point. If so where is the point.
99 Sample Problems Section If X is on the bisector of SKN, then X is equidistant from? and? 9. If X is equidistant from segments SK and SN, then X lies on the? 11. If O is on the perpendicular bisector of segment AF, then O is equidistant from? and? F S X K O N L A
100 Sample Problems Section Given: P is on the perpendicular bisector of P is on the perpendicular bisector of Prove: PA = PC AB BC A B C P
101 Sample Problems Section Prove theorem 4-6 Given: AB = AC Prove: A is on the perpendicular bisector of BC A B 1 2 X C
102 Sample Problems Section Prove theorem 4-8 Given: PX BA; PY BC; PX = PY Prove: BP bisects ABC X A P B C Y
103 Sample Problems Section a. A town wants to build a beach house on the lakefront equidistant from the recreation center and the school. Copy the diagram and show the point B where the beach house should be located. 19b. The town also wants to build a boat-launching site that is equidistant from Elm Road and Main Street. Find the point L where it should be. 19c. On your diagram, locate the point F for a flagpole that is to be the same distance from the recreation center, the school, and the courthouse.
104 Elm Road Sample Problems Section 4-7 lake courthouse recreation center school Main Street
105 Sample Problems Section Given: AB AC; BD AC; CE AB Prove: BD CE A E D B C
106 Sample Problems Section Given: SR is the perpendicular bisector of QT QR is the perpendicular bisector of SP Prove: PQ = TS Q S P R T
107 Sample Problems Section Given: Plane M is the perpendicular bisector of AB. Prove: AD BD; AC BC; CAD CBD
108 Chapter Four Congruent Triangles Review Homework Page 162: 2-18 evens
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