Chapter 4: Congruent Triangles Objectives: ­Identify the corresponding parts of congruent figures. ­Prove 2 triangles congruent by using, SSS Postulate, SAS Postulate, and ASA Postulate. ­Deduce information about segments and angles after proving that 2 triangles are congruent. 4.1 Congruent Figures ­whenever 2 figures have the same size and shape, they are called congruent. ­You are told that ABC DEF so the following occurs: Corresponding angles Corresponding sides A↔D AB↔DE B↔E BC↔EF C↔F AC↔DF Congruent triangles are defined as: 2 triangles are congruent if and only if their vertices can be matched up so that the corresponding parts (angles and sides) of the triangle are congruent. S
R U A N Y When referring to congruent triangles, we name their corresponding vertices in the same order. The following statements about these triangles are also correct, since corresponding vertices of the triangles are named in the same order. Suppose you are given that XYZ ABC. From the definition of congruent triangles you know, for example, that XY AB and X A. When the definition of congruent triangles is used to justify either of these statements, the wording commonly used is, “corresponding parts of congruent triangles are congruent” often written CPCTC. 2 polygons are congruent if and only if their vertices can be matched up so that their corresponding parts are congruent. For example, B C E R H K A BRAKE O CHOKE Notice that side KE of pentagon BRAKE corresponds to side KE of pentagon CHOKE. KE is called a common side of the 2 pentagons. Examples: Suppose IM
TIM BER. Complete the following: Angle R MTI If ABC XYZ, measure of angle B =80, and measure of angle C=50, name 4 congruent angles. P X Y T
Triangle PXY Triangle _______ Angle P Angle____ because ___________________________. XP _____ because ___________________________. Angle 1 Angle 2 because ____________________________. Then YX bisects Angle PYT because ____________________________________. HW pp. 120­121 #2­20 evens. 4.2 Some Ways to Prove Triangles Congruent Postulate 12 (SSS Postulate) If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent. B 3 G 5 4 5 4 3 A F C H Postulate 13 (SAS Postulate) If 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent. O B G L D 3 M 9 9 A 60° 3 60° F Y E C H Postulate 14 (ASA Postulate) If 2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent. M B H 4 F N O 55°
A C G 55° 4 Y K E Examples B A D C A B X C Z Y Y Given: BA YZ; BA bisects angle YZB Prove: Triangle AYB congruent to Triangle AZB A Z Statements 1. BA 2. B Reasons YZ; BA bisects angle YZB 3. Angle 3 congruent angle 4 4. AB congruent AB 5. Homework pp. 124­126 # 2­18 evens
1. 2. If 2lines are perpendicular, they form congruent adjacent angles. 3. 4. 5. ASA Postulate 4.3 Using Congruent Triangles Objective: To deduce information about segments or angles once we’ve shown that they are corresponding parts of congruent triangles. Some proofs require the idea of a line perpendicular to a plane. A line and a plane are perpendicular if and only if they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection.. A Way to Prove 2 Segments or 2 Angles Congruent: ­Identify 2 triangles in which the 2 segments or angles are corresponding parts. ­Prove that the triangles are congruent. ­State that the 2 parts are congruent, using the reason: CPCTC (Corr. parts of congruent triangles are congruent.) Example Given: m 1 = m 2; m 3 = m 4 Prove: M is the midpt of JK Statements 1. 2. 3. 4. 5. 6. Given: MK = OK; KJ bisects MKO Prove: JK bisects MJO Statements 1. MK = OK; KJ bisects MKO 2. 3 = 4 3. JK = JK 4. MJK = OJK 5. 1 = 2 6. JK bisects MJO Reasons 1. 2. 3. 4. 5. 6. Reasons 1. 2. 3. 4. 5. 6.
Given: P = S; O is the midpt of PS. Prove: O is the midpt of RQ. Statements 1. P = S; O is the midpt of PS 2. PO = SO 3. POQ = SOR 4. POQ = SOR 5. QO = RO 6. O is the midpt of RQ Homework pp. 130­131 # 2­12 even
Reasons 1. 2. 3. 4. 5. 6. 4.4 The Isosceles Triangle Theorems Theorem 4.1 (The Isosceles Triangle Theorem) If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. Corollary 1 An equilateral triangle is also equiangular. Corollary 2 An equilateral triangle has (3) 60° angles. Corollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpt. Theorem 4.2 (Converse Isosceles Triangle Theorem) The 2 angles of a triangle are congruent, then the sides opposite those angles are congruent. Corollary (Converse Corollary 1) An equiangular triangle is also equilateral. Example BC = AC 1. If m 1 = 140, m 2 = ___, m 3 = ___, m 4 = ___. 2. If the m 4 = 65, m 3 = ___, m 2 = ___, m 1= ___.
Given: j || k; AB = CB Prove: m 1 = m 2 Statements 1. 2. 3. 4. 5. 6. Reasons 1. 2. 3. 4. 5. 6. Find the value of x. Homework pp. 137­139 # 2­10 even, 14,18, 22­30 even
4.5 Other Methods of Proving Triangles Congruent. Theorem 4.3 (AAS Theorem) If 2 angles and a non­included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. Theorem 4.4 (HL Theorem) 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. Summary of Ways to Prove Triangles Congruent: All triangles: SSS, SAS, ASA, AAS Right triangles: HL Example Given: XY AB ; XA = XB Prove: 1 = 2 Statements 1. XY AB; XA = XB 2. 3 and 4 are right angles. 3. AYX and BYX are right triangles. 4. XY = XY 5. AYX = ______ 6. Given: Prove: Statements 1. 2. 3. 4. Reasons 1. 2. 3. 4. 5. 6. Reasons 1. 2. 3. 4.
Find the value of x. Homework pp. 143­145 # 2­14 even
4.6 Using More than One Pair of Congruent Triangles Given: Prove: Statements 1. 2. 3. 4. 5. 6. 7. Reasons 1. 2. 3. 4. 5. 6. 7. Reason for each step of the proof that Triangle BOX Triangle DOY Triangle ABX Triangle CDY
4.7 Medians, Altitudes, and Perpendicular Bisectors Median: a segment from the vertex to the midpoint of the opposite side. Altitude: the perpendicular segment from the vertex to the line that contains the opposite side. In a right triangle, 2 of the altitudes are the legs of the triangle, and the 3 rd one is inside. In an obtuse triangle, 2 of the altitudes are on the outside of the figure, and the 3 rd one inside. Perpendicular bisector: a segment is a line ( or ray or segment) that is perpendicular to the segment at its midpoint. Theorem 4­5 If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.
Theorem 4­6 If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. Theorem 4­7 If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem 4­8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. If Z lies on the bisector of , then Z is equidistant from ______ and _______. If P is equidistant from CA and CB, then P lies on __________________________. Given a. b. c. d. RST and a point K, it is known that KR=5, KS=5, and K lies on the bisector of Can RST be: An equilateral triangle? An isosceles, not equilateral, triangle? An acute scalene triangle? A right triangle? Homework pp 156­157 #2, 4, 7­12, 19­20