Math-2 Lesson 8-6 Unit 5 review -midpoint, -distance, -angles, -Parallel lines, -triangle congruence -triangle similarity -properties of

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1 Math- Lesson 8-6 Unit 5 review -midpoint, -distance, -angles, -Parallel lines, -triangle congruence -triangle similarity -properties of parallelograms -properties of Isosceles triangles

2 The distance between any two numbers: d a b between -5 and 3 ( 5) (3) between -10 and 3 between 7 and -4 8 ( 10) (3) ( 7) ( 4)

3 Vocabulary Theorem is a statement that has been proven to be true. Theorems are usually written in IF _hypothesis, THEN _conclusion format. Meaning : If the hypothesis is true then we know the conclusion is true.

4 The Pythagorean Theorem: If the triangle is a right triangle, Then the lengths of the sides are related by: a b c

5 What is the distance between (0,0) and (4, 3)? AB = 5 AB =? BC = 3 AC = 4 a b c (3) (4) c a b c 9 16 c 5 c

6 Vocabulary The midpoint of a segment is the point whose distance is half-way between the endpoints of the segment. We can find the midpoint between any two numbers using the following formula a b

7 We can find the midpoint of a segment that is on the (x, y) plane using the following formula: x 1 x y1 y, If we have points A and B on the (x, y) plane then we could use a more-easily understood formula: A X B X, A Y B Y

8 Midpoint of AB =? 1, A X B 1, X, A Y B 4 1 3, Y

9 Your Turn D 1) D is the included angle of which two sides? DF and DE ) What is the included angle of sides DF and EF? F 3) DF is the included side of which two angles? D and F 4) What is the included side of D and E DE E F

10 Angles can be represented three ways. (1) Using an angle symbol followed by one letter representing the vertex of the angle. We can use this only if there is only one angle with that vertex. A () Using an angle symbol followed by three letters representing a point on one side of the angle, the vertex, and a point on the other side of the angle. 3 BAC CAB 3 (3) Using an angle symbol followed by one number that is a label and NOT the measure of the angle.

11 Your turn: 3 (1) Can be named? If not, why can t it be? () Represent two other ways. 3 ABC CBA B No because we don t know which angle it is we just say. B

12 Vocabulary Included side: If two angles in a triangle are given, the included side is the side that is between the two angles or side that both of the angles have in common. RS is the included side of R and S What is the included Side for S and T? ST is the included side of S and T

13 Vocabulary Included angle: If two sides of a triangle are given, the included angle is the angle formed by those two sides. T is the included angle of RT and TS What is the included angle of SR and RT? R is the included angle of SR and RT

14 Your Turn Each pair of triangles is congruent: Write a congruence statement for each pair of triangle. ΔDEF ΔPRQ ΔRST ΔGFH

15 What triangle congruence theorem can we use to prove the two triangles are congruent? Angle-Side-Angle (ASA) Congruency Axiom: if two angles and their included side are congruent, then the two triangles are congruent. ABC DEG BC EG BCA EGD Therefore, ΔABC ΔDEG by ASA

16 What triangle congruence theorem can we use to prove the two triangles are congruent? Side-Side-Side (SSS) Congruency Theorem: if all three pairs of corresponding sides of a triangle are congruent, then the triangles are congruent AB DE BC EF CA FD Therefore, ΔABC ΔDEF by SSS

17 What triangle congruence theorem can we use to prove the two triangles are congruent? Side-Angle-Side (SAS) Congruency Theorem: if two pairs of corresponding sides and the pair of included angles are congruent, then the triangles are congruent. XZ QR ZXY RQF XY QF Therefore, ΔXYZ ΔQFR by SAS

18 What triangle congruence theorem can we use to prove the two triangles are congruent? Angle-Angle-Side (AAS) Congruency Theorem: If two pairs of corresponding angles are congruent and one pair of corresponding sides are congruent (which are NOT the included side), then the two triangles are congruent. ZXY EFD XYZ FDE XZ FE Therefore, ΔXYZ ΔFDE by AAS

19 Side-Side-Angle (SSA) Condition Let s look at an example of ASS A D AB DE BC EF

20 The Triangle Sum Theorem If the polygon is a triangle then the sum of the interior angles = 180 m A + m B + m C = 180

21 The Triangle Sum Theorem If the polygon is a triangle then the sum of the interior angles = 180 m A + m B + m C = x =? 3x 5 = x - 5 3x = x = x = 30 x = 10

22 If two parallel lines are cut by a transversal then: congruent. Alternate Interior Angles are congruent. congruent. Alternate Exterior Angles are Corresponding Angles are Consecutive Interior Angles are supplementary

23 If two lines intersect then: congruent. Vertical Angles are supplementary Linear Pair Angles are 1 3 4

24 What sequence of angles would you link to prove m4 m :Alternate Interior Angle Theorem 5 9 : Corresponding Angles Thm 4 9 :Substitution

25 Diagonal: a segment that connects opposite vertices of 4-sided polygons. A 1 B m1 m Because: D C AB is parallel to CD BD Is a transversal. 1 and are alternate interior angles.

26 Given that figure ABCD is a parallelogram, prove that ABD CDB 1 ma BD DB 1 mc alternate interior angles Opposite angles of a parallelogram (anything is congruent to itself) ABD CDB AAS congruence Parallelogram Properties : 3. Diagonals form congruent triangles. Theorem

27 What can we say about the measures of? 1 A B and CD AB CD Why? ABD CDB If triangles are congruent then all 6 corresponding pairs of angles and sides are congruent (CPCTC) Similarly: AD CB

28 Properties of Parallelograms congruent. Opposite Angles are supplementary congruent. Adjacent Angles are Opposite Sides are

29 Properties of Parallelograms Two congruent triangles. Diagonals form Diagonals form Congruent Alt Int. Angles

30 Properties of Parallelograms Two pairs of congruent triangles. Two Diagonals form Diagonals Bisect each other

31 Given: Segment AM is an angle bisector of vertex angle A. What additional congruence statement can we write based upon the given information? CAM BAM Prove AC CAM BAM AB CAM BAM AM AM CAM and BAM are included angles by definition given AC is anangle bisector same segment by definition CAM BAM SAS Theorem C A M B

32 1. Base Angles are congruent.. The vertex angle bisector: a. Forms two congruent triangles. 3 Properties of Isosceles Triangles Resulting equations : mc mb * mcam mcab A b. Is a perpendicular bisector of the base. mcma mbma 90 mcma mbma 180 CM MB CB *MB C M B

33 Angle-Angle (AA) Triangle Similarity: If two pairs of corresponding angles are congruent then the triangles are similar. G A E B

34 Side-Side-Side (SSS) Triangle Similarity: If all three pairs of corresponding sides are proportional then the triangles are similar. AB GE BC EF AC GF

35 Side-Angle-Side (SAS) Triangle Similarity: IF two pairs of corresponding sides are proportional and the included angles are congruent THEN the triangles are similar. G A AB GE AC GF

36 Scale Factor: the number that is multiplied by the length of each side of one triangle to equal the lengths of the sides of the other triangle. GE(scale factor) scale factor GEFABC AB AB GE Scale factor 3

37 Draw a Constructing a Perpendicular Bisector line segment. Draw two arcs of equal radius. Construct a point where the arcs intersect.