1. Use the figure to answer the questions.
1. What are the values of x and y?
ANSWER
125, 125
2. If AX and BY intersect at point P, what kind of triangle is XPY?
ANSWER
isosceles

2. Classify quadrilaterals by their properties.
Target
Classify quadrilaterals by their properties.
You will…
Use properties of trapezoids and kites.

3. Vocabulary
B and C
trapezoid – a quadrilateral with exactly one pair of parallel sides;
parallel sides are called bases
non-parallel sides are legs
two base angle pairs
A and D
isosceles trapezoid – a trapezoid with congruent legs

4. Vocabulary
Isosceles Trapezoids
Isosceles Trapezoid Theorem 8.14 – If a trapezoid is isosceles, then each pair of base angles is congruent
Isosceles Trapezoid Converse Theorem 8.15 – If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
Trapezoid Diagonals Theorem 8.16 – A trapezoid is isosceles if and only if its diagonals are congruent.

5. Vocabulary
Midsegments
midsegment of a trapezoid – the segment that connects the midpoints of its legs
Midsegment Theorem for Trapezoids –
the midsegment of a trapezoid is
1) parallel to both bases and
2) its length is the average of the bases
MN = AB + DC = (AB + DC) 2 1

6. Vocabulary
Kites
kite – a quadrilateral that has two pair of congruent consecutive sides, but opposite sides are not congruent
Kite Theorem 8.18 – The diagonals of a kite are perpendicular.
Kite Theorem 8.19 – Exactly one pair of opposite angles of a kite are congruent.

7. EXAMPLE 1
Use a coordinate plane
Show that ORST is a trapezoid.
SOLUTION
Compare the slopes of opposite sides.
4 – 3
2 – 0 = 1 2
Slope of RS =
so RS OT
2 – 0
4 – 0 = 2 4 1
Slope of OT =
2 – 4
4 – 2 = –2 2 –1
Slope of ST =
so ST is not parallel to OR 3
3 – 0
0 – 0 =
undefined
Slope of OR =
Because quadrilateral ORST has exactly one pair of parallel sides, it is a trapezoid.
ANSWER

8. GUIDED PRACTICE
for Example 1
Suppose the coordinates of a quadrilateral are R(0, 3), S(4, 5), T(4, 2), O(0, 0). What type of quadrilateral is ORST? Explain.
ANSWER
Parallelogram; opposite pairs of sides are parallel.
In Exercise 1, which of the interior angles of quadrilateral ORST are supplementary angles? Explain your reasoning. 2.
ANSWER
O and R , T and S; Consecutive Interior Angles Theorem

9. EXAMPLE 2
Use properties of isosceles trapezoids
Arch
The stone above the arch in the diagram is an isosceles trapezoid. Find m K, m M, and m J.
SOLUTION
STEP 1
Find m K. JKLM is an isosceles trapezoid, so K and L are congruent base angles, and m K = m L= 85o.

10. EXAMPLE 2
Use properties of isosceles trapezoids
STEP 2
Find m M. Because L and M are consecutive interior angles formed by LM intersecting two parallel lines, they are supplementary. So, m M = 180o – 85o = 95o.
STEP 3
Find m J. Because J and M are a pair of base angles, they are congruent, and m J = m M = 95o.
ANSWER
So, m J = 95o, m K = 85o, and m M = 95o.

11. Use the midsegment of a trapezoid
EXAMPLE 3
Use the midsegment of a trapezoid
In the diagram, MN is the midsegment of trapezoid PQRS. Find MN.
SOLUTION
Use Theorem 8.17 to find MN.
MN (PQ + SR) 1 2 =
Apply Theorem 8.17.
= ( ) 1 2
Substitute 12 for PQ and 28 for XU.
= 20
Simplify.
ANSWER
The length MN is 20 inches.

12. GUIDED PRACTICE
for Examples 2 and 3
In Exercises 3 and 4, use the diagram of trapezoid EFGH.
3. If EG = FH, is trapezoid EFGH isosceles? Explain.
ANSWER
yes, Theorem 8.16
4. If m HEF = 70o and m FGH = 110o, is trapezoid EFGH isosceles? Explain.
SAMPLE ANSWER
Yes;
m EFG = 70° by Consecutive Interior Angles Theorem making EFGH an isosceles trapezoid by Theorem 8.15.

13. 15 cm; Solve for x to find ML.
GUIDED PRACTICE
for Examples 2 and 3
5. In trapezoid JKLM, J and M are right angles, and JK = 9 cm. The length of the midsegment NP of trapezoid JKLM is 12 cm. Sketch trapezoid JKLM and its midsegment. Find ML. Explain your reasoning. J L K M
9 cm
12 cm N P
ANSWER
( 9 + x ) = 12 1 2
15 cm; Solve for x to find ML.

14. Find m D in the kite shown at the right.
EXAMPLE 4
Apply Theorem 8.19
Find m D in the kite shown at the right.
SOLUTION
By Theorem 8.19, DEFG has exactly one pair of congruent opposite angles. Because E G, D and F must be congruent. So, m D = m F. Write and solve an equation to find m D.
m D + m F +124o + 80o = 360o
Corollary to Theorem 8.1
m D + m D +124o + 80o = 360o
Substitute m D for m F.
2(m D) +204o = 360o
Combine like terms.
m D = 78o
Solve for m D.

15. GUIDED PRACTICE
for Example 4
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?
ANSWER
25; 75o