6-5 Conditions for Special Parallelograms Warm Up Lesson Presentation Lesson Quiz Holt Geometry

Warm Up 5 –1 1. Find AB for A (–3, 5) and B (1, 2). 2. Find the slope of JK for J(–4, 4) and K(3, –3). ABCD is a parallelogram. Justify each statement. 3. ABC  CDA 4. AEB  CED 5 –1  opp. s  Vert. s Thm.

Objective Prove that a given quadrilateral is a rectangle, rhombus, or square.

When you are given a parallelogram with certain properties, you can use the theorems below to determine whether the parallelogram is a rectangle.

Example 1: Carpentry Application A manufacture builds a mold for a desktop so that , , and mABC = 90°. Why must ABCD be a rectangle? Both pairs of opposites sides of ABCD are congruent, so ABCD is a . Since mABC = 90°, one angle ABCD is a right angle. ABCD is a rectangle by Theorem 6-5-1.

Below are some conditions you can use to determine whether a parallelogram is a rhombus.

In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram. Caution To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43.

You can also prove that a given quadrilateral is a rectangle, rhombus, or square by using the definitions of the special quadrilaterals. Remember!

Example 3A: Identifying Special Parallelograms in the Coordinate Plane Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–1, 4), Q(2, 6), R(4, 3), S(1, 1)

Example 3A Continued Step 1 Graph PQRS.

Example 3A Continued Step 2 Find PR and QS to determine is PQRS is a rectangle. Since , the diagonals are congruent. PQRS is a rectangle.

Step 3 Determine if PQRS is a rhombus. Example 3A Continued Step 3 Determine if PQRS is a rhombus. Since , PQRS is a rhombus. Step 4 Determine if PQRS is a square. Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.

Example 3B: Identifying Special Parallelograms in the Coordinate Plane Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. W(0, 1), X(4, 2), Y(3, –2), Z(–1, –3) Step 1 Graph WXYZ.

Example 3B Continued Step 2 Find WY and XZ to determine is WXYZ is a rectangle. Since , WXYZ is not a rectangle. Thus WXYZ is not a square.

Example 3B Continued Step 3 Determine if WXYZ is a rhombus. Since (–1)(1) = –1, , PQRS is a rhombus.

Check It Out! Example 3B Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–4, 6) , Q(2, 5) , R(3, –1) , S(–3, 0)

Check It Out! Example 3B Continued Step 1 Graph PQRS.

Check It Out! Example 3B Continued Step 2 Find PR and QS to determine is PQRS is a rectangle. Since , PQRS is not a rectangle. Thus PQRS is not a square.

Check It Out! Example 3B Continued Step 3 Determine if KLMN is a rhombus. Since (–1)(1) = –1, are perpendicular and congruent. KLMN is a rhombus.

Objectives Use properties of kites to solve problems. Use properties of trapezoids to solve problems.

Vocabulary kite trapezoid base of a trapezoid leg of a trapezoid base angle of a trapezoid isosceles trapezoid midsegment of a trapezoid

A kite is a quadrilateral with exactly two pairs of congruent consecutive sides.

Example 1: Problem-Solving Application Lucy is framing a kite with wooden dowels. She uses two dowels that measure 18 cm, one dowel that measures 30 cm, and two dowels that measure 27 cm. To complete the kite, she needs a dowel to place along . She has a dowel that is 36 cm long. About how much wood will she have left after cutting the last dowel?

Understand the Problem Example 1 Continued 1 Understand the Problem The answer will be the amount of wood Lucy has left after cutting the dowel. 2 Make a Plan The diagonals of a kite are perpendicular, so the four triangles are right triangles. Let N represent the intersection of the diagonals. Use the Pythagorean Theorem and the properties of kites to find , and . Add these lengths to find the length of .

Example 1 Continued Solve 3 N bisects JM. Pythagorean Thm. Pythagorean Thm.

Example 1 Continued Lucy needs to cut the dowel to be 32.4 cm long. The amount of wood that will remain after the cut is, 36 – 32.4  3.6 cm Lucy will have 3.6 cm of wood left over after the cut.

Example 1 Continued 4 Look Back To estimate the length of the diagonal, change the side length into decimals and round. , and . The length of the diagonal is approximately 10 + 22 = 32. So the wood remaining is approximately 36 – 32 = 4. So 3.6 is a reasonable answer.

Check It Out! Example 1 What if...? Daryl is going to make a kite by doubling all the measures in the kite. What is the total amount of binding needed to cover the edges of his kite? How many packages of binding must Daryl buy?

Understand the Problem Check It Out! Example 1 Continued 1 Understand the Problem The answer has two parts. • the total length of binding Daryl needs • the number of packages of binding Daryl must buy

Check It Out! Example 1 Continued 2 Make a Plan The diagonals of a kite are perpendicular, so the four triangles are right triangles. Use the Pythagorean Theorem and the properties of kites to find the unknown side lengths. Add these lengths to find the perimeter of the kite.

Check It Out! Example 1 Continued Solve 3 Pyth. Thm. Pyth. Thm. perimeter of PQRS =

Check It Out! Example 1 Continued Daryl needs approximately 191.3 inches of binding. One package of binding contains 2 yards, or 72 inches. packages of binding In order to have enough, Daryl must buy 3 packages of binding.

Check It Out! Example 1 Continued 4 Look Back To estimate the perimeter, change the side lengths into decimals and round. , and . The perimeter of the kite is approximately 2(54) + 2 (41) = 190. So 191.3 is a reasonable answer.

A trapezoid is a quadrilateral with exactly one pair of parallel sides A trapezoid is a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base angles of a trapezoid are two consecutive angles whose common side is a base.

If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.

Theorem 6-6-5 is a biconditional statement Theorem 6-6-5 is a biconditional statement. So it is true both “forward” and “backward.” Reading Math

Example 3A: Using Properties of Isosceles Trapezoids Find mA. mC + mB = 180° Same-Side Int. s Thm. 100 + mB = 180 Substitute 100 for mC. mB = 80° Subtract 100 from both sides. A  B Isos. trap. s base  mA = mB Def. of  s mA = 80° Substitute 80 for mB

Check It Out! Example 3a Find mF. mF + mE = 180° Same-Side Int. s Thm. E  H Isos. trap. s base  mE = mH Def. of  s mF + 49° = 180° Substitute 49 for mE. mF = 131° Simplify.

Check It Out! Example 3b JN = 10.6, and NL = 14.8. Find KM. Isos. trap. s base  KM = JL Def. of  segs. JL = JN + NL Segment Add Postulate KM = JN + NL Substitute. KM = 10.6 + 14.8 = 25.4 Substitute and simplify.

Example 4A: Applying Conditions for Isosceles Trapezoids Find the value of a so that PQRS is isosceles. Trap. with pair base s   isosc. trap. S  P mS = mP Def. of  s Substitute 2a2 – 54 for mS and a2 + 27 for mP. 2a2 – 54 = a2 + 27 Subtract a2 from both sides and add 54 to both sides. a2 = 81 a = 9 or a = –9 Find the square root of both sides.

Example 4B: Applying Conditions for Isosceles Trapezoids AD = 12x – 11, and BC = 9x – 2. Find the value of x so that ABCD is isosceles. Diags.   isosc. trap. Def. of  segs. AD = BC Substitute 12x – 11 for AD and 9x – 2 for BC. 12x – 11 = 9x – 2 Subtract 9x from both sides and add 11 to both sides. 3x = 9 x = 3 Divide both sides by 3.

Check It Out! Example 4 Find the value of x so that PQST is isosceles. Trap. with pair base s   isosc. trap. Q  S mQ = mS Def. of  s Substitute 2x2 + 19 for mQ and 4x2 – 13 for mS. 2x2 + 19 = 4x2 – 13 Subtract 2x2 and add 13 to both sides. 32 = 2x2 Divide by 2 and simplify. x = 4 or x = –4

The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it.

Example 5: Finding Lengths Using Midsegments Find EF. Trap. Midsegment Thm. Substitute the given values. EF = 10.75 Solve.

Substitute the given values. Check It Out! Example 5 Find EH. Trap. Midsegment Thm. 1 16.5 = (25 + EH) 2 Substitute the given values. Simplify. 33 = 25 + EH Multiply both sides by 2. 13 = EH Subtract 25 from both sides.

Lesson Quiz: Part I 1. Erin is making a kite based on the pattern below. About how much binding does Erin need to cover the edges of the kite? In kite HJKL, mKLP = 72°, and mHJP = 49.5°. Find each measure. 2. mLHJ 3. mPKL about 191.2 in. 81° 18°

Lesson Quiz: Part II Use the diagram for Items 4 and 5. 4. mWZY = 61°. Find mWXY. 5. XV = 4.6, and WY = 14.2. Find VZ. 6. Find LP. 119° 9.6 18