Math Geometry Chapter 8 Test Review Key

Chapter 6 Test Review Geometry CP 1. Find the sum of the measures of a 25-gon. (25-2) * 180 = 4140° 2. The sum of the measures of the interior angles of a convex polygon is 6480°. Classify the polygon by its number of sides. (n-2) * 180 = 6480 n = 38 3. Find the measures of an interior angle and an exterior angle of a regular heptagon. Interior angle: (7-2) * 180 / 7 = 128.6° Exterior angle: 360 / 7 = 51.4° 4. Is it possible for a regular polygon to have an interior angle measure of 160°? Why or why not? Yes, it's possible. 160 = (n-2) * 180 n = 18 (18-gon) 5. Find the value of x. 5x + 7x + 10x + 9x + 1 + 9x + 1 + 10x + 5x = 900 x = 16 6. Find the value of the variables in the parallelograms. a) 3j = 5j - 9, j = 9/2 b) k + 10 = 6k, k = 2 7. 12j - 41 + 5j - 17 = 180 j = 14 13k - 9 = 5k + 7 k = 2 8. 3x² + 4x - 14 = 2x² + 6x + 1 x² - 2x - 15 = 0 (x - 5)(x + 3) = 0 x = 5, -3 9. 3x - 4 = 5y + 1 5x - 4 = 3y + 4 Solving the system of equations: x = 5, y = 2 10. 3x² + 4x - 14 = 2x² + 6x + 1 x² - 2x - 15 = 0 (x - 5)(x + 3) = 0 x = 5, -3 11. ABCD is a parallelogram. Find x, y, and z. x = 11 y = 8 z = 120° 12. Find m∠B in the picture, ABCD is a parallelogram. Given: ∠A = 50°, ∠C = 34° Solution: ∠B = 180° - (∠A + ∠C) = 180° - (50° + 34°) = 96° For any rhombus, ABCD, decide whether the statement is always, sometimes, or never true. Justify your reasoning. (Draw a picture) ∠A ≅ ∠C Always true. Opposite angles in a rhombus are congruent. DA = AB Always true. All sides of a rhombus are congruent. AC = BD Sometimes true. The diagonals of a rhombus are only congruent if the rhombus is also a square. ∠B + ∠C = 90° Never true. The sum of adjacent angles in a rhombus is 180°, not 90°. For any rectangle, FGHJ, decide whether the statement is always, sometimes, or never true. Justify your reasoning. (Draw a picture) 1   1. ∠G ≅ ∠H Always true. All angles in a rectangle are right angles. JF = FG Sometimes true. This is only true if the rectangle is also a square. Classify the parallelogram. Explain your reasoning. If necessary find the value of any variables. Rhombus: All sides are congruent. Rectangle: Opposite sides are congruent and one angle is a right angle. Square: All sides are congruent and one angle is a right angle. Rhombus: All sides are congruent. 6y - 5 = 3y + 7 3y = 12 y = 4 2x - 11 + 127 = 180 2x = 54 x = 27 Given rhombus RQST and its given values (m∠QPR = 30° and RT = 6), find the measures below: m∠QTP = 90° RP = 6√3 Solution: Using the 30-60-90 triangle properties, RP = 2 * RT = 2 * 6 = 12 Then, using the Pythagorean theorem on triangle QPR, we get RP = 6√3 QT = 6√3 Similar to RP, QT = 2 * RT = 2 * 6 = 12 Using the Pythagorean theorem on triangle QRT, we get QT = 6√3 Find the side lengths of the kite. Write the lengths in simplest radical form. (Hint: What do you know about the diagonals of a kite?) x² + 4² = y² 2² + 4² = x² x = 2√5, y = 4√5 4² + 8² = y² y = 4√5 Find the value of x. 12 = (16 + 4x) / 2 x = 2 20.5 = (27 + 6x - 1) / 2 x = 2.5 WEST is a kite. Find the missing angle measures. m∠WT = 113° m∠EW = 97° m∠E = 118° m∠T = 118° Points A, B, C, and D are the vertices of a quadrilateral. Determine whether ABCD is a trapezoid. A(-2, 3), B(3, 3), C(-1, -2), D(2, -2) AB = 5, CD = 5, AC = 5, BD = 5 AB || CD, but AC ≠ BD Therefore, ABCD is a trapezoid. Problem 34: Given: Coordinates of points P(-2, 4), Q(4, 4), R(2, -1), and S(-4, -1) Task: Determine the most specific name for the quadrilateral. Solution: Calculate slopes: PQ: 0 QR: -5 SR: 0 PS: -5 Calculate distances: PQ = 6 QR = √10 SR = 6 PS = √10 Since opposite sides are parallel and congruent, the quadrilateral is a parallelogram. Problem 35: Given: Coordinates of points P(-1, 2), Q(4, 4), R(1, -2), and S(-4, -4) Task: Determine the most specific name for the quadrilateral. Solution: Calculate slopes: PQ = 2/5 QR = -3 SR = 2/5 PQ = -3 Calculate distances: PQ = √29 QR = 3 SR = √29 PQ = 3 Since two pairs of adjacent sides are congruent but opposite sides are not, the quadrilateral is a kite. Problem 36: Determine whether each statement is always, sometimes, or never true. a. Diagonals of a trapezoid are congruent. Sometimes b. Opposite sides of a rectangle are congruent. Always c. A square is a rectangle. Always   d. A square is not a rhombus. Never e. All angles of a parallelogram are congruent. Sometimes f. The diagonals of a parallelogram are perpendicular. Sometimes   Problem 37: Given: ECTA is a rectangle with EC = 48, ER = 6x - 14, CA = 50, CT = 14, and EA = 5y + 4. Task: Find x and y. Solution: 6x - 14 = 25, x = 6.5 5y + 4 = 14, y = 2 Problem 38: Given: RECT is a rectangle with ∠T = 2x + y + 6 and ∠C = x + 2y - 6. Task: Find x and y. Solution: x + 2y - 6 = 90 2x + y + 6 = 90 Solving the system of equations, x = 24 and y = 36 Problem 39: Given: Quadrilateral ABCD with AB ≅ CD and AB || CD. Task: Give the most descriptive name for the quadrilateral. Solution: Parallelogram