Geometry Chapter 6 Concept Review
Angles of Polygons: The sum of the measure of the inter angles of a polygon is given by the formula S=180(n-2) where n represents the number of sides/angles. The sum of the measures of the exterior angles of a convex polygon is 360. To find the measure of one angle divide the sum by n (the number of sides or angles). When you are given a quadrilateral and asked to solve for x in the expressions given for angles, find the sum of the interior angles then add all the expressions and set it equal to the sum. Solve for x.
Properties of Parallelograms: Opposite sides are congruent and parallel. Opposite angles are congruent. Consecutive angles are supplementary (both add to equal 180). If a parallelogram has one right angle, it has four right angles. Diagonals bisect each other. Remember that alternate interior angles are congruent.
Test for Parallelograms: If a quadrilateral has the properties listed above for a parallelogram, then it is a parallelogram.
Properties of Rectangles, Rhombi, Squares, and Trapezoids: A rectangle has all the properties of a parallelogram. Diagonals are congruent and bisect each other. All four angles are right angles. A rhombus has all the properties of a parallelogram. All sides are congruent. Diagonals are perpendicular. Each diagonal bisects a pair of opposite angles. To prove diagonals of a rhombus are perpendicular, find the slopes of the diagonals. A square has all the properties of a parallelogram, a rectangle, and a rhombus. A Trapezoid is a quadrilateral with exactly one pair of parallel sides. In an isosceles trapezoid, the legs are congruent, both pairs of base angles are congruent and the diagonals are congruent. Trapezoid: Median = (Base + Base)/2
Always draw a picture and graph it to help you understand the problem!!! Practice Problems:
1. Find the sum of the measures of the interior angles of a convex 30-gon.
S=180(n-2) S=180(30-2) S=180(28) S=5040 degrees
2. Find the sum of the measure of the exterior angles of a 28-gon.
360
3. Find the measure of an exterior angle of a 28-gon.
360/28 = 12.86
4. Solve for x.
S=180(n-2) S=180(5-2) S=180(3) S=540
Add all the interior angles together. 5x +30 = 540 5x = 510 X=102
5. For parallelogram ABCD, find each measure: m BCD, segment AF, m BDC, and segment BC.
m BCD = 20 + 32 = 52 AF = 6.86 m BDC = 180-52-40.1 = 87.9 BC = 9 6. Which of the following is a property of all parallelograms? a. Each pair of opposite angles is congruent. b. Only one pair of opposite sides is congruent. c. Each pair of opposite angles is supplementary. d. There are four right angles.
7. ABCD is a parallelogram with vertices H(0,4), J(-4,6), and K(5,6). Find the fourth coordinate L.
L(9,4) or L(-9,4) or L(1,8)
8. Which of the following is true for all rectangles? a. The diagonals are perpendicular. b. The diagonals bisect the angles. c. The consecutive sides are congruent. d. The consecutive sides are perpendicular.
9. What would you do to prove that the diagonals of a rectangle are congruent?
Find the length of the diagonals. If the lengths are equal, then the diagonals are congruent.
10. ABCD is a rectangle. If AB = 2x + 1, and CD = 3x - 25, find x.
Draw the rectangle and label the vertices. Opposite sides are congruent. 2x + 1 = 3x – 25 1 = 1x – 25 26 = 1x X = 26
11.How would you prove that the diagonals of a rhombus are perpendicular?
Find the slopes of the diagonals. If they are opposite sign, reciprocal then the diagonals are perpendicular.
12.For the rhombus below, find m KMJ.
Alternate interior angles are congruent. m KMJ = 28 degrees
13.The diagonals of square ABCD intersect at E. If AE = 7x – 6 and BD = 5x + 30, find AC.
Diagonals of a square bisect each other. Diagonals of a square are congruent. 7x – 6 + 7x -6 = 5x + 30 14x -12 = 5x + 30 9x = 42 X = 4.67
AC = 5(4.67) +30 = 53.35
14.The length of the base of a trapezoid is 20 inches and the length of median is 30 inches. Find the length of the other base.
Median = (Base + Base)/2 30 = (20 + Base)/2 2(30) = 2(20 + Base)/2 60 = 20 + Base Base = 40
15.In this isosceles trapezoid, find m X.
Isosceles Trapezoids have congruent base angles. m X = 70
HONORS GEOMETRY ONLY:
1. Name the missing coordinates for the following quadrilateral.
A(-b,b) E(b,-b)