Math-2 Lesson 5-4 Parallelograms and their Properties Isosceles Triangles and Their Properties
Segment Bisector: A point on the interior of a segment that is the midpoint of the segment. This midpoint can be the point of intersection of the segment with a) Another segment b) A ray c) A line. M This would make the other segment, ray, or line the segment bisector
Constructing a Perpendicular Bisector 1. Draw a line segment.
Constructing a Perpendicular Bisector 1. Draw a line segment. 2. Draw two arcs of equal radius. 3. Construct a point where the arcs intersect.
Constructing a Perpendicular Bisector 4. Draw a line through the two points.
Segment EF is a perpendicular bisector of segment AB. Are there any equations that we can write from this result? make mbke 90 AK BK AB 2* AK
Angle Bisector a common side of two angles that divides the angle into two angles of equal measure. B A 1 2 D Common Side THEN of. C If m1 m2 BC ABD is an angle bisector Are there any equations that we can write from this result? mabc m ABC 2* mdbc mdbc
One pair of parallel lines 8 5 7 3 6 1 Name 4 2 Name Name an alternate Interior angle to 6 an alternate Exterior angle to 3: 2 : 8 Name a corresponding angle to 3: 8 Name a Consecutive Interior angle to 3: 5 Name an angle vertical to 3: an angle that forms a 2 linear pair with angle 3: 1 or 4
Two pairs of parallel lines 3 1 4 2 13 14 16 15 8 5 7 6 9 10 12 11
What sequence of angles would you link to prove 1 2 13 14 3 4 16 15 m4 m9 8 5 7 6 9 10 12 11 4 5:Alternate Interior Angle Theorem 5 9 : Corresponding Angles Thm 4 9 :Substitution
Parallelogram: a 4-sided polygon whose opposite sides are parallel. 1 3 2 4 Parallelogram Properties : 13 14 1. Opposite Angles 16 15 are congruent. m4 m9 5 8 7 6 9 12 10 11 m6 m16
Parallelogram: a 4-sided polygon whose opposite sides are parallel. 1. Parallelogram Properties : Opposite Angles are congruent. ma mb mc md 2. Consecutive Interior Angles are supplementary. ma mb 180
Diagonal: a segment that connects opposite vertices of 4-sided polygons. BD Is a diagonal.
Diagonal: a segment that connects opposite vertices of 4-sided polygons. 1 How do the measure of 1 and 2 compare? 2 m1 m2 Why?
Diagonal: a segment that connects opposite vertices of 4-sided polygons. A 1 B m1 m2 Because: D 2 C BD AB is parallel to CD Is a transversal. 1 and 2 are alternate interior angles.
Given that figure ABCD is a parallelogram, prove that ABD CDB 2 1 ma BD DB ABD CDB Parallelogram Properties : 3. 1 2 alternate interior angles mc Opposite angles of a parallelogram (anything is congruent to itself) AAS congruence Theorem Diagonals form congruent triangles.
What can we say about the measures of? A B and CD 1 AB CD Why? 2 ABD CDB If triangles are congruent then all 6 corresponding pairs of angles and sides are congruent (CPCTC) Similarly: AD C B
What can we say about the opposite sides of parallelograms? 1 Parallelogram Properties : 4. Opposite sides are congruent. AB CD 2 AD CB
What can you say about the measures of and? A B 3 4 m3 m4 3 Because: D 4 C AD is parallel to CB BD Is a transversal. 3 and 4 are alternate interior angles.
Construct diagonal AC. What congruence statements can we make so far? 1 3 m1 m2 m3 m4 AB CD 4 AD CB 2
1 3 m1 m2 m3 m4 AB CD 4 AD CB 2 What last two steps do we need to prove AMD CMB? mamd mcmb Vertical Angle Theorem
What congruence statements can we make so far? 1 3 m1 m2 m3 m4 AB CD 4 AD CB 2 AMD CMB How do the measures of A M and CM compare? A M CM Why? (CPCTC)
What can we say about the diagonals of parallelograms? 1 3 4 2 5. Parallelogram Properties : Diagonals bisect each other. A M CM D M BM
Parallelogram Properties : 1. 2. 4 Opposite Angles are congruent. Consecutive Interior Angles are supplementary. 3 m3 m4 1 2 m1 m2 m3 180 AB CD 3. Diagonals form congruent triangles. 4. Opposite sides are congruent. 5. Diagonals bisect each other. Resulting equations : AM MC AC 2*MC
Isosceles Triangle: A triangle with two congruent sides. Vertex Angle: (Of an Isosceles Triangle) The included angle of the legs. Legs: (Of an Isosceles Triangle) The two congruent sides. Base Angles: (Of an Isosceles Triangle) The angle that includes the base. Base: (Of an Isosceles Triangle) The opposite the vertex angle.
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 same segment bisector by definition CAM BAM SAS Theorem C A M B
Given: Segment AM is an angle bisector of vertex angle A. We have proven this results in 2 congruent triangles. CAM BAM 1. Prove CM BM CPCTC 2. Prove mcma mbma CPCTC 3. Prove macm mabm CPCTC A C M B
Given: Segment AM is an angle bisector of vertex angle A. We have proven this results in 2 congruent triangles. 1. Prove CAM BAM CM BM CPCTC 2. Prove mcma mbma CPCTC A 3. Prove mcma mbma 90 Angle CMB is a straight angle. mcmb 180 mcma mbma mcma mcma mcma 90 By definition 180 180 Property of Equality By definition C Angle Addition Postulate Substitution (Theorem 2 above) M B
3 Properties of Isosceles Triangles 1. Base Angles are congruent. 2. The vertex angle bisector: a. Forms two congruent triangles. b. Is a perpendicular bisector of the base. Resulting equations : mc mb 2* mcam mcab A mcma mbma 90 mcma mbma 180 CM MB CB 2*MB C M B