Math-2. Lesson 7-4 Properties of Parallelograms And Isosceles Triangles
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1 Math-2 Lesson 7-4 Properties of Parallelograms nd Isosceles Triangles
2 What sequence of angles would you link to prove m4 m lternate Interior Corresponding
3 What sequence of angles would you link to prove: m 4 + m 6 = Corresponding Linear Pair
4 The two red lines are parallel, what can you say about... Linear ngle Pairs: supplementary Consecutive Interior ngles supplementary Vertical angle pair: congruent Corresponding ngles: congruent lternate Interior ngles: congruent lternate Exterior ngles: congruent
5 Parallelogram Properties : 1. Opposite ngles are congruent. m mb mc md 2. Consecutive Interior ngles are supplementary. m mb 180
6 Math Problems from Opposite ngles of Parallelograms are Congruent B 4x 50 C x =? C 2x + 10 m = m C 2x + 10 = 4x 50 D m =? x = 30 m = 2x + 10 m = m = 70
7 Math Problems from djacent ngles of Parallelograms are Supplementary B 4x 70 C x =? m D + m C = 180 2x + 10 D 2x x 70 = 180 6x = 240
8 Math Problems from djacent ngles of Parallelograms are Supplementary B 3x 1 3x Segment C is a diagonal. BC DC 2x + 6 m CD + m DC + m D = 180 3x 1 + 2x = 180 D C lternate Interior ngles 5x = 180 x = 5 x =? m BC + m DC = m BCD ngle ddition Postulate m DC + m BCD = 180 djacent ngles of Parallelograms 3x 1 + 2x = 180 5x = 180 x = 5 Triangle ngle Sum Theorem (we ll prove this later).
9 If we could prove the diagonal forms two congruent triangles, we could use CPCTC to prove more properties of Parallelograms. 1 m mc Opposite ngles are congruent. 1 2 lternate Interior ngles 2 D = BC B = CD CPCTC CPCTC BD = DB Same segment same length BC CDB S Theorem
10 Math Problems from Opposite Sides of Parallelograms are congruent 2x + 10 B D C 4x 70 B =? B = 2x + 10 B = 2(40) + 10 B = 90 x =? B = CD 2x + 10 = 4x 70 2x = 80 x = 40
11 Can we prove that diagonals form two pairs of congruent triangles? D CB Opposite Sides are congruent. MD CMB Vertical ngles 3 4 lternate Interior ngles 2 Using the other pairs of: 1) Opposite sides 2) Vertical angles 3) lternate Interior ngles CMD MB S Theorem MD CMB S Theorem
12 By CPCTC 1 3 DM MB M CM 4 2 Therefore, diagonals of parallelograms bisect each other.
13 Math Problems from Diagonals of Parallelograms BISECT each other. B M 2. Write an equation that relates the lengths in the problem. 2 M = C 2 3x 5 = Solve for x. 3x 5 = 13 3x = 18 x = 6 D C C = 26 M = 3x 5 x =? 1. Draw a picture of the diagonal and label the known measurements. 3x 5 26 M C
14 Parallelogram Properties : 1. Opposite ngles are congruent. m3 m4 2. Consecutive Interior ngles are supplementary. m1 m2 m diagonal of a parallelogram forms two congruent triangles. B CD DB CBD 4. Opposite Sides of parallelograms are congruent Opposite triangles formed by the diagonals (plural) form congruent triangles. MD CMB 6. Diagonals of parallelograms bisect each other. M MC C 2* MC
15 Segment Bisector: if a line segment is intersected by a ray, segment or line at the midpoint of the segment, then the ray, segment line is a segment bisector. a) nother segment b) ray c) line. M
16 EF is a perpendicular bisector of B. re there any equations (that come from congruencies) that we can write from this result? mke mbke 90 perpendicular bisector K BK perpendicular bisector B 2* K segment addition
17 Math Problems from Perpendicular Bisectors B M D 2. Write an equation that relates the lengths in the problem. 2 M = C 2 3x 5 = Solve for x. 3x 5 = 13 3x = 18 x = 6 C C = 26 M = 3x 5 3x 5 x =? 1. Draw a picture of the segment and label the known measurements. 26 M C
18 ngle Bisector: a common side of two adjacent angles that divides the angle into two angles of equal measure. B 1 2 D Common Side THEN of. C If m1 m2 BC BD is an angle bisector re there any equations that we can write from this result? mbc angle bisector m BD 2* angle bisector mdbc mdbc
19 Math Problems from ngle Bisectors B 76 3x + 1 D C BD is an angle bisector of x =? m BD = m CBD 3x + 1 =76 3x = 75 x = 25 BC To solve for an unknown value, you need an equation.
20 Isosceles Triangle: triangle with two congruent sides. Legs: (Of an Isosceles Triangle) The two congruent sides. Vertex ngle: (Of an Isosceles Triangle) The included angle of the legs. Base: (Of an Isosceles Triangle) The opposite the vertex angle. Base ngles: (Of an Isosceles Triangle) The angles that include the base.
21 ΔBC M Given: is an Isosceles Triangle and is an angle bisector of vertex angle. What other congruencies result from this statement? CM BM C B ( M bisects ) (Isosceles Triangle) M M CM BM (congruent to itself) (SS) C M B
22 CM BM Congruent triangles give us SIX Pairs of congruencies. CM BM mcm mbm mcm mbm C M B
23 Properties of Isosceles Triangles 1. The vertex and bisector forms two congruent triangles. CM BM 2. The vertex angle bisector is a perpendicular bisector of the base. m CM = m BM = 90 CM BM 3. Base ngles are congruent. mcm mbm C M B
24 Triangle Sum Theorem: If, B, and C are the interior angles of a triangle, then their measures add up to Line f m 1 + m 2 + m 3 = Line f Line k m 4 + m 2 + m 5 = 180 Line k Line k Line f Line k Line f m 1 m 4 = 3 5 m 3 m 5 =
25 Math Problems from The Triangle Sum Theorem. 1. Write an equation that relates the measures of the angles. m + m B + m C = Substitute the measures of the angles into the equation. 2x 1 + 3x x + 3 = Solve for x. 9x + 9 = 180 9x = 171 x = 19 m B =? m B = 4x + 3 m B = 4(19) + 3 m B = 79 C 3x + 7 2x 1 4x + 3 B
26 Constructing a Perpendicular Bisector Given a line segment B 1) Using a compass draw two arcs of equal radius using the endpoints as the center of each are. 2) Construct a point where the two arcs intersect. M 3) Construct a line through these two points. EF 4) Is the perpendicular bisector of B
27 Constructing an ngle Bisector Given B 1) Using a compass draw an arc using point B as the center. 2) Construct two points (points and C) where the arc intersects the side of the angles B C 3) Construct C 4) Construct a perpendicular bisector of C 5) BM is the angle bisector of BC M
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