TrapezoidsTrapezoids 5-5

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

EXAMPLE 1 Use a coordinate plane Slope of ST = 2 – 4 4 – 2 = –2 2 = –1 Slope of OR = – 0 0 – 0 =, which is undefined The slopes of ST and OR are not the same, so ST is not parallel to OR. Because quadrilateral ORST has exactly one pair of parallel sides, it is a trapezoid. ANSWER

GUIDED PRACTICE for Example 1 1. What If? In Example 1, suppose the coordinates of point S are (4, 5). What type of quadrilateral is ORST ? Explain. Compare the slopes of opposite sides. SOLUTION Slope of RS = 5 – 3 4 – 0 = 1 2 Slope of OT = 2 – 0 4 – 0 = 1 2 The slopes of RS and OT are the same, so RS OT.

GUIDED PRACTICE for Example 1 Slope of ST = Slope of OR = – 0 0 – 0 =, undefined The slopes of ST and OR are the same, so ST is parallel to OR. 2 – 5 4 – 4 = –3 0 undefined ANSWER Parallelogram; opposite pairs of sides are parallel.

GUIDED PRACTICE for Example 1 In Example 1, which of the interior angles of quadrilateral ORST are supplementary angles? Explain your reasoning. 2. 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

EXAMPLE 4 Apply Theorem 8.19 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. Find m D in the kite shown at the right.

m D + m F +124 o + 80 o = 360 o Corollary to Theorem 8.1 m D + m D +124 o + 80 o = 360 o 2(m D) +204 o = 360 o Combine like terms. Substitute m D for m F. Solve for m D. m D = 78 o EXAMPLE 4 Apply Theorem 8.19

GUIDED PRACTICE for Example 4 6. In a kite, the measures of the angles are 3x o, 75 o, 90 o, and 120 o. Find the value of x. What are the measures of the angles that are congruent? STEP 1 3x = 360° 3x3x = 75 x = 25 Sum of the angles in a quadrilateral = 360° 3x = 360° Subtract Divided by 3 from each side Combine like terms SOLUTION

GUIDED PRACTICE for Example 4 STEP 2 3x3x = 3 25 = 75 Substitute Simplify The value of x is 25 and the measures of the angles that are congruent is 75 ANSWER

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= 85 o.

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 = 180 o – 85 o = 95 o. STEP 3 Find m J. Because J and M are a pair of base angles, they are congruent, and m J = m M =95 o. ANSWER So, m J = 95 o, m K = 85 o, and m M = 95 o.

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

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, 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.

GUIDED PRACTICE for Examples 2 and 3 4. If m HEF = 70 o and m FGH = 110 o, is trapezoid EFGH isosceles?Explain. 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 ANSWER G and F are consecutive interior angles as EF HG. Because FGH=110°, therefore EFG=70° as they are supplementary angles.

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. ANSWER NP is the midsegment of trapezoid JKLM. and NP = ( JK + ML) 1 2 J L K M 9 cm 12 cm x cm N P

GUIDED PRACTICE for Examples 2 and 3 Apply Theorem Substitute Simplify. = (9 + x) = 9 + x 24 = x 15 Multiply each side by 2 (JK + ML) 1 2 = NP