Transcript Slide 1
Properties of Kites 6-6 of Kites and Trapezoids 6-6 Properties and Trapezoids Holt Holt Geometry Geometry 6-6 Properties of Kites and Trapezoids Warm Up Solve for x. 1. x2 + 38 = 3x2 – 12 5 or –5 2. 137 + x = 180 43 3. 156 4. Find FE. Holt Geometry 6-6 Properties of Kites and Trapezoids A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. Holt Geometry 6-6 Properties of Kites and Trapezoids Holt Geometry 6-6 Properties of Kites and Trapezoids 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? Holt Geometry 6-6 Properties of Kites and Trapezoids Example 2A: Using Properties of Kites In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mBCD. Kite cons. sides ∆BCD is isos. 2 sides isos. ∆ CBF CDF isos. ∆ base s mCBF = mCDF Def. of s mBCD + mCBF + mCDF = 180° Polygon Sum Thm. Holt Geometry 6-6 Properties of Kites and Trapezoids Example 2A Continued mBCD + mCBF + mCDF = 180° Substitute mCDF mBCD + mCBF + mCDF = 180° for mCBF. mBCD + 52° + 52° = 180° mBCD = 76° Holt Geometry Substitute 52 for mCBF. Subtract 104 from both sides. 6-6 Properties of Kites and Trapezoids Example 2B: Using Properties of Kites In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mABC. ADC ABC Kite one pair opp. s Def. of s Polygon Sum Thm. mABC + mBCD + mADC + mDAB = 360° mADC = mABC Substitute mABC for mADC. mABC + mBCD + mABC + mDAB = 360° Holt Geometry 6-6 Properties of Kites and Trapezoids Example 2B Continued mABC + mBCD + mABC + mDAB = 360° mABC + 76° + mABC + 54° = 360° 2mABC = 230° mABC = 115° Holt Geometry Substitute. Simplify. Solve. 6-6 Properties of Kites and Trapezoids Example 2C: Using Properties of Kites In kite ABCD, mDAB = 54°, and mCDF = 52°. Find mFDA. CDA ABC Kite one pair opp. s mCDA = mABC Def. of s mCDF + mFDA = mABC Add. Post. 52° + mFDA = 115° mFDA = 63° Holt Geometry Substitute. Solve. 6-6 Properties of Kites and Trapezoids 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. Holt Geometry 6-6 Properties of Kites and Trapezoids If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid. Holt Geometry 6-6 Properties of Kites and Trapezoids Holt Geometry 6-6 Properties of Kites and Trapezoids Reading Math Theorem 6-6-5 is a biconditional statement. So it is true both “forward” and “backward.” Holt Geometry 6-6 Properties of Kites and Trapezoids Example 3A: Using Properties of Isosceles Trapezoids Find mA. mC + mB = 180° 100 + mB = 180 Same-Side Int. s Thm. Substitute 100 for mC. mB = 80° A B Subtract 100 from both sides. Isos. trap. s base mA = mB Def. of s mA = 80° Substitute 80 for mB Holt Geometry 6-6 Properties of Kites and Trapezoids Example 3B: Using Properties of Isosceles Trapezoids KB = 21.9m and MF = 32.7. Find FB. Isos. trap. s base KJ = FM Def. of segs. KJ = 32.7 Substitute 32.7 for FM. KB + BJ = KJ Seg. Add. Post. 21.9 + BJ = 32.7 Substitute 21.9 for KB and 32.7 for KJ. BJ = 10.8 Subtract 21.9 from both sides. Holt Geometry 6-6 Properties of Kites and Trapezoids Example 3B Continued Same line. KFJ MJF Isos. trap. s base Isos. trap. legs ∆FKJ ∆JMF SAS BKF BMJ CPCTC FBK JBM Vert. s Holt Geometry 6-6 Properties of Kites and Trapezoids Example 3B Continued Isos. trap. legs ∆FBK ∆JBM AAS CPCTC Holt Geometry FB = JB Def. of segs. FB = 10.8 Substitute 10.8 for JB. 6-6 Properties of Kites and Trapezoids Check It Out! Example 3a Find mF. mF + mE = 180° E H mE = mH mF + 49° = 180° mF = 131° Holt Geometry Same-Side Int. s Thm. Isos. trap. s base Def. of s Substitute 49 for mE. Simplify. 6-6 Properties of Kites and Trapezoids Check It Out! Example 3b JN = 10.6, and NL = 14.8. Find KM. Isos. trap. s base KM = JL JL = JN + NL Def. of segs. KM = JN + NL Substitute. Segment Add Postulate KM = 10.6 + 14.8 = 25.4 Substitute and simplify. Holt Geometry 6-6 Properties of Kites and Trapezoids 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 mS = mP 2a2 – 54 = a2 a2 Substitute 2a2 – 54 for mS and + 27 2 a + 27 for mP. = 81 a = 9 or a = –9 Holt Geometry Def. of s Subtract a2 from both sides and add 54 to both sides. Find the square root of both sides. 6-6 Properties of Kites and Trapezoids 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. AD = BC Def. of segs. Substitute 12x – 11 for AD and 12x – 11 = 9x – 2 9x – 2 for BC. 3x = 9 x=3 Holt Geometry Subtract 9x from both sides and add 11 to both sides. Divide both sides by 3. 6-6 Properties of Kites and Trapezoids Check It Out! Example 4 Find the value of x so that PQST is isosceles. Q S mQ = mS Trap. with pair base s isosc. trap. Def. of s 2 + 19 for mQ Substitute 2x 2x2 + 19 = 4x2 – 13 and 4x2 – 13 for mS. 32 = 2x2 x = 4 or x = –4 Holt Geometry Subtract 2x2 and add 13 to both sides. Divide by 2 and simplify. 6-6 Properties of Kites and Trapezoids 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. Holt Geometry 6-6 Properties of Kites and Trapezoids Holt Geometry 6-6 Properties of Kites and Trapezoids Example 5: Finding Lengths Using Midsegments Find EF. Trap. Midsegment Thm. Substitute the given values. EF = 10.75 Holt Geometry Solve. 6-6 Properties of Kites and Trapezoids Check It Out! Example 5 Find EH. Trap. Midsegment Thm. 1 16.5 = 2 (25 + EH) Substitute the given values. Simplify. 33 = 25 + EH Multiply both sides by 2. 13 = EH Subtract 25 from both sides. Holt Geometry