B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure...
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Transcript of B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure...
![Page 1: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/1.jpg)
B
D
AC
Conjecture: mB = 2(mA)
7.
![Page 2: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/2.jpg)
A
B
CD
E
30
x
BCAB
BEBD
1.
From HW # 6
Given:
x = 15
Find the measure of the angle marked x.
![Page 3: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/3.jpg)
x
2.
x = 15
From HW # 6
![Page 4: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/4.jpg)
A
C
B
D
63
63
41104
![Page 5: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/5.jpg)
A
C
B
D
63
104
63
41
![Page 6: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/6.jpg)
A
C
B
D
63
104
![Page 7: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/7.jpg)
A
C
B
D
63
104 76
63
41
D
![Page 8: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/8.jpg)
A B
C
D E F
14
![Page 9: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/9.jpg)
5. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
C
D F
A E B
From HW # 6
![Page 10: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/10.jpg)
C
D F
A E B
Outline of proof:
1. A B (isosceles triangle theorem)
From HW # 65. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
![Page 11: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/11.jpg)
C
D F
A E B
Outline of proof:
1. A B (isosceles triangle theorem)2. AE EB (definition of midpoint)
From HW # 65. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
![Page 12: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/12.jpg)
C
D F
A E B
Outline of proof:
1. A B (isosceles triangle theorem)2. AE EB (definition of midpoint)3. ADE and BFE are congruent right angles
From HW # 65. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
![Page 13: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/13.jpg)
C
D F
A E B
Outline of proof:
1. A B (isosceles triangle theorem)2. AE EB (definition of midpoint)3. ADE and BFE are congruent right angles4. ADE BFE (AAS)
From HW # 65. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
![Page 14: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/14.jpg)
C
D F
A E B
Outline of proof:
1. A B (isosceles triangle theorem)2. AE EB (definition of midpoint)3. ADE and BFE are congruent right angles4. ADE BFE (AAS)5. DE EF and AD BF (CPCTC)
From HW # 65. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
![Page 15: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/15.jpg)
C
D F
A E B
Outline of proof:
1. A B (isosceles triangle theorem)2. AE EB (definition of midpoint)3. ADE and BFE are congruent right angles4. ADE BFE (AAS)5. DE EF and AD BF (CPCTC)6. DEF is isosceles (definition of isosceles )
From HW # 65. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
![Page 16: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/16.jpg)
C
D F
A E B
Outline of proof:
1. A B (isosceles triangle theorem)2. AE EB (definition of midpoint)3. ADE and BFE are congruent right angles4. ADE BFE (AAS)5. DE EF and AD BF (CPCTC)6. DEF is isosceles (definition of isosceles ) 7. DC FC (subtraction post, AC – AD = BC – BF)
From HW # 65. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
![Page 17: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/17.jpg)
C
D F
A E B
Outline of proof:
1. A B (isosceles triangle theorem)2. AE EB (definition of midpoint)3. ADE and BFE are congruent right angles4. ADE BFE (AAS)5. DE EF and AD BF (CPCTC)6. DEF is isosceles (definition of isosceles ) 7. DC FC (subtraction post, AC – AD = BC – BF)8. CDF is isosceles.
From HW # 65. In the diagram, triangle ABC is isosceles with base , E is the midpoint of , and . Prove that triangle DEF and triangle CDF are isosceles.
ABBCEF,ACED AB
![Page 18: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/18.jpg)
D E
C
B
A
6. Think about how you would prove that the altitudes to the legs of an isosceles triangle are congruent.
From HW # 6
![Page 19: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/19.jpg)
B
D
AC
Conjecture: mB = 2(mA)
7.
![Page 20: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/20.jpg)
Use Geometer’s Sketchpad to construct quadrilateral ABCD in which AB is parallel to CD and BC is parallel to AD.
DC
B A
![Page 21: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/21.jpg)
Quadrilaterals
![Page 22: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/22.jpg)
DC
B A
A rhombus is a parallelogram with one pair of adjacent sides congruent.
DC
B A
A rectangle is a parallelogram with one right angle.
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
DC
B A
DC
B A
A square is a rhombus and a rectangle.
Definitions
![Page 23: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/23.jpg)
B A
C D
A trapezoid is a quadrilateral with exactly one pair of sides parallel.
An isosceles trapezoid is a trapezoid with the two non-parallel sides congruent.
B A
C D
bases
legs
Base angles (there are two pairs)
![Page 24: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/24.jpg)
Parallelogram Rhombus Rectangle Square Trapezoid Isosceles Trapezoid
Exactly 1 pair of parallel sides
2 pairs of parallel sides
Exactly 1 pair of congruent sides
2 pairs of congruent sides
All sides congruent
Perpendicular diagonals
Congruent diagonals
Diagonals bisect angles
Diagonals bisect each other
Opposite angles congruent
Four right angles
Base angles congruent
![Page 25: B D A C Conjecture: m B = 2(m A) 7.. A B C D E 30 x 1. From HW # 6 Given: x = 15 Find the measure of the angle marked x.](https://reader035.fdocuments.in/reader035/viewer/2022062517/56649e745503460f94b73f1e/html5/thumbnails/25.jpg)
HW #7
Fill in the chart (question 5) on the HW 7 handout.
Begin work on the rest of the problems.
You will have all period on Monday to complete it.