Congruent Triangles Triangles. Warm Up Lesson Presentation Lesson Quiz. Holt Geometry. McDougal Geometry

Size: px

Start display at page:

Download "Congruent Triangles Triangles. Warm Up Lesson Presentation Lesson Quiz. Holt Geometry. McDougal Geometry"

May O’Connor’
5 years ago
Views:

1 Triangles Warm Up Lesson Presentation Lesson Quiz Holt Geometry McDougal Geometry

2 Warm Up 1. Name all sides and angles of FGH. FG, GH, FH, F, G, H 2. What is true about K and L? Why? ;Third s Thm. 3. What does it mean for two segments to be congruent? They have the same length.

3 Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence.

4 Vocabulary corresponding angles corresponding sides congruent polygons

5 Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

7 Helpful Hint Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices.

8 To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence statement, the order of the vertices indicates the corresponding parts.

9 Helpful Hint When you write a statement such as ABC DEF, you are also stating which parts are congruent.

10 Example 1: Naming Congruent Corresponding Parts Given: PQR STW Identify all pairs of corresponding congruent parts. Angles: P S, Q T, R W Sides: PQ ST, QR TW, PR SW

11 Check It Out! Example 1 If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts. Angles: L E, M F, N G, P H Sides: LM EF, MN FG, NP GH, LP EH

12 Example 2A: Using Corresponding Parts of Congruent Triangles Given: ABC DBC. Find the value of x. BCA and BCD are rt. s. BCA BCD m BCA = m BCD (2x 16) = 90 2x = 106 x = 53 Def. of lines. Rt. Thm. Def. of s Substitute values for m BCA and m BCD. Add 16 to both sides. Divide both sides by 2.

13 Example 2B: Using Corresponding Parts of Congruent Triangles Given: ABC DBC. Find m DBC. m ABC + m BCA + m A = 180 m ABC = 180 m ABC = 180 Sum Thm. Substitute values for m BCA and m A. Simplify. m ABC = 40.7 DBC ABC m DBC = m ABC Subtract from both sides. Corr. s of s are. Def. of s. m DBC 40.7 Trans. Prop. of =

14 Check It Out! Example 2a Given: ABC DEF Find the value of x. AB DE AB = DE 2x 2 = 6 2x = 8 x = 4 Corr. sides of s are. Def. of parts. Substitute values for AB and DE. Add 2 to both sides. Divide both sides by 2.

15 Check It Out! Example 2b Given: ABC DEF Find m F. m EFD + m DEF + m FDE = 180 ABC DEF m ABC = m DEF Sum Thm. Corr. s of are. Def. of s. m DEF = 53 Transitive Prop. of =. m EFD = 180 m F = 180 m F = 37 Substitute values for m DEF and m FDE. Simplify. Subtract 143 from both sides.

16 Example 3: Proving Triangles Congruent Given: YWX and YWZ are right angles. YW bisects XYZ. W is the midpoint of XZ. XY YZ. Prove: XYW ZYW

17 Statements 1. YWX and YWZ are rt. s. 2. YWX YWZ 3. YW bisects XYZ Reasons 1. Given 2. Rt. Thm. 3. Given 4. XYW ZYW 4. Def. of bisector 5. W is mdpt. of XZ 6. XW ZW 7. YW YW 8. X Z 9. XY YZ 10. XYW ZYW 5. Given 6. Def. of mdpt. 7. Reflex. Prop. of 8. Third s Thm. 9. Given 10. Def. of

18 Check It Out! Example 3 Given: AD bisects BE. BE bisects AD. AB DE, A D Prove: ABC DEC

19 Statements 1. A D 2. BCA DCE 3. ABC DEC 4. AB DE 5. AD bisects BE, BE bisects AD 6. BC EC, AC DC 7. ABC DEC Reasons 1. Given 2. Vertical s are. 3. Third s Thm. 4. Given 5. Given 6. Def. of bisector 7. Def. of s

20 Example 4: Engineering Application The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. Given: PR and QT bisect each other. PQS RTS, QP RT Prove: QPS TRS

21 Example 4 Continued Statements 1. QP RT 2. PQS RTS 3. PR and QT bisect each other. 4. QS TS, PS RS 5. QSP TSR 6. QSP TRS 7. QPS TRS Reasons 1. Given 2. Given 3. Given 4. Def. of bisector 5. Vert. s Thm. 6. Third s Thm. 7. Def. of s

22 Check It Out! Example 4 Use the diagram to prove the following. Given: MK bisects JL. JL bisects MK. JK ML. JK ML. Prove: JKN LMN

23 Check It Out! Example 4 Continued Statements 1. JK ML 2. JK ML 3. JKN NML 4. JL and MK bisect each other. 5. JN LN, MN KN 6. KNJ MNL 7. KJN MLN 8. JKN LMN Reasons 1. Given 2. Given 3. Alt int. s are. 4. Given 5. Def. of bisector 6. Vert. s Thm. 7. Third s Thm. 8. Def. of s

24 Lesson Quiz 1. ABC JKL and AB = 2x JK = 4x 50. Find x and AB. 31, 74 Given that polygon MNOP polygon QRST, identify the congruent corresponding part. 2. NO RS 3. T P 4. Given: C is the midpoint of BD and AE. A E, AB ED Prove: ABC EDC

25 Lesson Quiz 4. Statements 1. A E 2. C is mdpt. of BD and AE 3. AC EC; BC DC 4. AB ED 5. ACB ECD 6. B D 7. ABC EDC Reasons 1. Given 2. Given 3. Def. of mdpt. 4. Given 5. Vert. s Thm. 6. Third s Thm. 7. Def. of s

26 HOMEWORK: Pg #3-10, 23-25

TEKS: G10B, G9B, G5B, G2B The student will justify and apply triangle congruence relationships. The student will formulate and test conjectures about

TEKS: G10B, G9B, G5B, G2B The student will justify and apply triangle congruence relationships. The student will formulate and test conjectures about the properties and attributes of polygons and their