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Trapezoids. 5-5. Show that ORST is a trapezoid. =. =. Slope of RS =. 4 – 3. 2 – 0. Slope of OT =. =. 2 – 0. 4 – 0. The slopes of RS and OT are the same, so RS OT. 2. 1. 1. 2. 4. 2. EXAMPLE 1. Use a coordinate plane. SOLUTION. Compare the slopes of opposite sides. –2.

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Presentation Transcript
slide2

Show that ORSTis a trapezoid.

=

=

Slope of RS =

4 – 3

2 – 0

Slope of OT =

=

2 – 0

4 – 0

The slopes of RSand OTare the same, so RSOT .

2

1

1

2

4

2

EXAMPLE 1

Use a coordinate plane

SOLUTION

Compare the slopes of opposite sides.

slide3

–2

–1

=

=

Slope of ST =

2

,

which is undefined

=

Slope of OR =

The slopes of ST and ORare not the same, soST is not parallel to OR .

2 – 4

3 – 0

ANSWER

0 – 0

4 – 2

Because quadrilateral ORST has exactly one pair of parallel sides, it is a trapezoid.

3

0

EXAMPLE 1

Use a coordinate plane

slide4

=

=

Slope of RS =

2 – 0

5 – 3

Slope of OT =

4 – 0

4 – 0

The slopes of RSand OTare the same, so RSOT .

1

1

2

2

for Example 1

GUIDED PRACTICE

1. What If?In Example 1, suppose the coordinates of point Sare (4, 5). What type of quadrilateral is ORST? Explain.

SOLUTION

Compare the slopes of opposite sides.

slide5

–3

undefined

=

Slope of ST =

0

,

undefined

=

Slope of OR =

The slopes of ST and ORare the same, soST is parallel to OR .

3 – 0

2 – 5

ANSWER

Parallelogram; opposite pairs of sides are parallel.

0 – 0

4 – 4

3

0

for Example 1

GUIDED PRACTICE

slide6

2.

In Example 1, which of the interior angles of quadrilateral ORSTare supplementary angles? Explain your reasoning.

ANSWER

O and R , T and S are supplementary angles, as RS and OR are parallel lines cut by transversals OR and ST, therefore the pairs of consecutive interior angles are supplementary by theorem 8.5

for Example 1

GUIDED PRACTICE

slide7

Find m Din the kite shown at the right.

SOLUTION

By Theorem 8.19, DEFGhas exactly one pair of congruent opposite angles. Because E G,Dand Fmust be congruent.So, m D = mF.Write and solve an equation to find mD.

EXAMPLE 4

Apply Theorem 8.19

slide8

mD + m F +124o + 80o = 360o

mD + m D +124o + 80o = 360o

Substitute m Dfor m F.

2(m D)+204o = 360o

m D = 78o

Solve for m D.

EXAMPLE 4

Apply Theorem 8.19

Corollary to Theorem 8.1

Combine like terms.

slide9

3x + 75 + 90 + 120

= 360°

3x + 285

= 360°

3x

= 75

x

= 25

for Example 4

GUIDED PRACTICE

6. In a kite, the measures of the angles are 3xo, 75o,90o, and 120o. Find the value of x. What are the measures of the angles that are congruent?

SOLUTION

STEP 1

Sum of the angles in a quadrilateral = 360°

Combine like terms

Subtract

Divided by 3 from each side

slide10

3x

ANSWER

The value of x is 25 and the measures of the angles that are congruent is 75

= 3 25

for Example 4

GUIDED PRACTICE

STEP 2

Substitute

= 75

Simplify

slide11

Arch

The stone above the arch in the diagram is an isosceles trapezoid. Find mK, mM, and mJ.

STEP 1

Find mK. JKLMis an isosceles trapezoid, so Kand Lare congruent base angles, and mK = mL= 85o.

EXAMPLE 2

Use properties of isosceles trapezoids

SOLUTION

slide12

STEP 2

Find m M. Because Land M are consecutive interior angles formed by LMintersecting two parallel lines,they are supplementary. So,mM = 180o – 85o = 95o.

STEP 3

Find mJ. Because J and M are a pair of base angles, they are congruent, and mJ = mM=95o.

ANSWER

So, mJ = 95o, m K = 85o, and m M = 95o.

EXAMPLE 2

Use properties of isosceles trapezoids

slide13

In the diagram,MNis the midsegment of trapezoidPQRS. FindMN.

SOLUTION

Use Theorem 8.17 to findMN.

1

= (12+ 28)

2

1

MN (PQ + SR)

=

2

ANSWER

The length MNis 20 inches.

EXAMPLE 3

Use the midsegment of a trapezoid

Apply Theorem 8.17.

Substitute 12 for PQand 28 for XU.

= 20

Simplify.

slide14

3. If EG = FH, is trapezoid EFGHisosceles? Explain.

ANSWER

Yes, trapezoid EFGH is isosceles, if and only if its diagonals are congruent. As, it is given its diagonals are congruent, therefore by theorem 8.16 the trapezoid is isosceles.

for Examples 2 and 3

GUIDED PRACTICE

In Exercises 3 and 4, use the diagram of trapezoidEFGH.

slide15

4. If mHEF = 70o and mFGH =110o, is trapezoid EFGHisosceles?Explain.

ANSWER

G and F are consecutive interior angles as EF HG. Because FGH=110°, therefore EFG=70° as they are supplementary angles.

BHG=110°, as the sum of a quadrilateral is 360° . The base angles are congruent that is, 110° each therefore, the trapezoid is isosceles by theorem 8.15.

for Examples 2 and 3

GUIDED PRACTICE

slide16

J

K

9 cm

12 cm

P

N

ANSWER

NPis the midsegment of trapezoidJKLM.

x cm

M

L

1

5. In trapezoid JKLM, Jand M are right angles, and JK = 9cm. The length of the midsegment NPof trapezoid JKLMis 12cm. Sketch trapezoid JKLMand its midsegment. Find ML. Explain your reasoning.

2

andNP = ( JK + ML)

for Examples 2 and 3

GUIDED PRACTICE

slide17

1

(JK + ML)

NP

=

2

12

24

= 9 + x

15

= x

1

= (9 + x)

2

for Examples 2 and 3

GUIDED PRACTICE

Apply Theorem 8.17.

Substitute

Multiply each side by 2

Simplify.