B

D

AC

Conjecture: mB = 2(mA)

7.

A

B

CD

E

30

x

BCAB

BEBD

1.

From HW # 6

Given:

x = 15

Find the measure of the angle marked x.

x

2.

x = 15

From HW # 6

A

C

B

D

63

63

41104

A

C

B

D

63

104

63

41

A

C

B

D

63

104

A

C

B

D

63

104 76

63

41

D

A B

C

D E F

14

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

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

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

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

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

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

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

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

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

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

B

D

AC

Conjecture: mB = 2(mA)

7.

Use Geometer’s Sketchpad to construct quadrilateral ABCD in which AB is parallel to CD and BC is parallel to AD.

DC

B A

Quadrilaterals

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

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)

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

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.