Splash Screen

Five-Minute Check (over Lesson 6–5) CCSS Then/Now New Vocabulary Theorems: Isosceles Trapezoids Proof: Part of Theorem 6.23 Example 1: Real-World Example: Use Properties of Isosceles Trapezoids Example 2: Isosceles Trapezoids and Coordinate Geometry Theorem 6.24: Trapezoid Midsegment Theorem Example 3: Standardized Test Example: Midsegment of a Trapezoid Theorems: Kites Example 4: Use Properties of Kites Lesson Menu

LMNO is a rhombus. Find x. A. 5 B. 7 C. 10 D. 12 5-Minute Check 1

LMNO is a rhombus. Find y. A. 6.75 B. 8.625 C. 10.5 D. 12 5-Minute Check 2

QRST is a square. Find n if mTQR = 8n + 8. B. 9 C. 8.375 D. 6.5 5-Minute Check 3

QRST is a square. Find w if QR = 5w + 4 and RS = 2(4w – 7). B. 5 C. 4 D. 3.3 _ 5-Minute Check 4

QRST is a square. Find QU if QS = 16t – 14 and QU = 6t + 11. B. 10 C. 54 D. 65 5-Minute Check 5

Which statement is true about the figure shown, whether it is a square or a rhombus? C. JM║LM D. 5-Minute Check 6

Mathematical Practices Content Standards G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. CCSS

midsegment of a trapezoid kite bases legs of a trapezoid base angles isosceles trapezoid midsegment of a trapezoid kite Vocabulary

The parallel sides of a trapezoid are called bases. A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs. The two angles that share a base of a trapezoid are called base angles. Trapezoids

An isosceles trapezoid is a trapezoid with congruent legs.

Isosceles Trapezoid Theorems Theorem: If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent. Isosceles Trapezoid Theorems

Theorem: If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

Theorem: If one pair of base angles of a trapezoid are congruent then the trapezoid is isosceles.

Use Properties of Isosceles Trapezoids A. BASKET Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and LN = 3.6 feet, find mMJK. Example 1A

Since JKLM is a trapezoid, JK║LM. Use Properties of Isosceles Trapezoids Since JKLM is a trapezoid, JK║LM. mJML + mMJK = 180 Consecutive Interior Angles Theorem 130 + mMJK = 180 Substitution mMJK = 50 Subtract 130 from each side. Answer: mMJK = 50 Example 1A

Use Properties of Isosceles Trapezoids B. BASKET Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and JL is 10.3 feet, find MN. Example 1B

JL = KM Definition of congruent JL = KN + MN Segment Addition Use Properties of Isosceles Trapezoids Since JKLM is an isosceles trapezoid, diagonals JL and KM are congruent. JL = KM Definition of congruent JL = KN + MN Segment Addition 10.3 = 6.7 + MN Substitution 3.6 = MN Subtract 6.7 from each side. Answer: MN = 3.6 Example 1B

A. Each side of the basket shown is an isosceles trapezoid A. Each side of the basket shown is an isosceles trapezoid. If mFGH = 124, FI = 9.8 feet, and IG = 4.3 feet, find mEFG. A. 124 B. 62 C. 56 D. 112 Example 1A

B. Each side of the basket shown is an isosceles trapezoid B. Each side of the basket shown is an isosceles trapezoid. If mFGH = 124, FI = 9.8 feet, and EG = 14.1 feet, find IH. A. 4.3 ft B. 8.6 ft C. 9.8 ft D. 14.1 ft Example 1B

Isosceles Trapezoids and Coordinate Geometry Quadrilateral ABCD has vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. A quadrilateral is a trapezoid if exactly one pair of opposite sides are parallel. Use the Slope Formula. Example 2

slope of slope of slope of Isosceles Trapezoids and Coordinate Geometry slope of slope of slope of Answer: Exactly one pair of opposite sides are parallel, So, ABCD is a trapezoid. Example 2

Use the Distance Formula to show that the legs are congruent. Isosceles Trapezoids and Coordinate Geometry Use the Distance Formula to show that the legs are congruent. Answer: Since the legs are not congruent, ABCD is not an isosceles trapezoid. Example 2

A. trapezoid; not isosceles B. trapezoid; isosceles C. not a trapezoid Quadrilateral QRST has vertices Q(–1, 0), R(2, 2), S(5, 0), and T(–1, –4). Determine whether QRST is a trapezoid and if so, determine whether it is an isosceles trapezoid. A. trapezoid; not isosceles B. trapezoid; isosceles C. not a trapezoid D. cannot be determined Example 2

The mid-segment of a trapezoid is the segment that joins the midpoints of its legs of a trapezoid. Midsegments

Theorem: If a quadrilateral is a trapezoid then, (1) the mid-segment is parallel to the bases and (2) the length of the mid-segment is half the sum (or the average) of the lengths of the bases.

Midsegment of a Trapezoid In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x? Example 3

WXYZ is an isosceles trapezoid with median Find XY if JK = 18 and WZ = 25. A. XY = 32 B. XY = 25 C. XY = 21.5 D. XY = 11 Example 3

Using the Midsegment of a Trapezoid QR is the mid-segment of trapezoid LMNP. What is x? Using the Midsegment of a Trapezoid

 MN is the midsegment of trapezoid PQRS. What is x? What is MN? MN is the mid-segment of trapezoid PQRS. What is the value of x? What is MN?  MN is the midsegment of trapezoid PQRS. What is x? What is MN?

Trapezoid Midsegment Theorem Midsegment of a Trapezoid Read the Test Item You are given the measure of the midsegment of a trapezoid and the measures of one of its bases. You are asked to find the measure of the other base. Solve the Test Item Trapezoid Midsegment Theorem Substitution Example 3

A kite is a quadrilateral with two pairs of consecutive sides that are congruent and NO opposite sides congruent.

Theorem: If a quadrilateral is a kite, then its diagonals are perpendicular. Segment AC is a bisector of segment BD. Kites

Theorem: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Finding Angle Measures in Kites Quadrilateral DEFG is a kite. What are m1, m2, and m3? Finding Angle Measures in Kites

A. If WXYZ is a kite, find mXYZ. Use Properties of Kites A. If WXYZ is a kite, find mXYZ. Example 4A

mW + mX + mY + mZ = 360 Polygon Interior Angles Sum Theorem Use Properties of Kites Since a kite only has one pair of congruent angles, which are between the two non-congruent sides, WXY  WZY. So, WZY = 121. mW + mX + mY + mZ = 360 Polygon Interior Angles Sum Theorem 73 + 121 + mY + 121 = 360 Substitution mY = 45 Simplify. Answer: mXYZ = 45 Example 4A

B. If MNPQ is a kite, find NP. Use Properties of Kites B. If MNPQ is a kite, find NP. Example 4B

NR2 + MR2 = MN2 Pythagorean Theorem (6)2 + (8)2 = MN2 Substitution Use Properties of Kites Since the diagonals of a kite are perpendicular, they divide MNPQ into four right triangles. Use the Pythagorean Theorem to find MN, the length of the hypotenuse of right ΔMNR. NR2 + MR2 = MN2 Pythagorean Theorem (6)2 + (8)2 = MN2 Substitution 36 + 64 = MN2 Simplify. 100 = MN2 Add. 10 = MN Take the square root of each side. Example 4B

Since MN  NP, MN = NP. By substitution, NP = 10. Use Properties of Kites Since MN  NP, MN = NP. By substitution, NP = 10. Answer: NP = 10 Example 4B

A. If BCDE is a kite, find mCDE. Example 4A

B. If JKLM is a kite, find KL. C. 7 D. 8 Example 4B

HOMEWORK PGS. 444-447 # 1,2,6-11,16-21,24-27,35,36 41-44

End of the Lesson