Proving Geometry Theorems Answers

University:
Winston-Salem Forsyth County Schools
Course:
AP Calculus AB
Academic year:

2014
Views:

405

Pages:

6
Author:

x3sweetheartz9gl

Math III Proving Geometric Theorems Review 1. Vertical Angles When two lines intersect, the angles across from each other are vertical angles and are congruent. 2. Parallel Lines Cut by a Transversal Congruent alternate interior and corresponding angles are produced: Corresponding Angles: 𝐴 = 𝐸 A=E, 𝐵 = 𝐹 B=F, 𝐶 = 𝐺 C=G, 𝐷 = 𝐻 D=H Alternate Interior Angles: 𝐶 = 𝐹 C=F, 𝐷 = 𝐸 D=E Alternate Exterior Angles: 𝐴 = 𝐻 A=H, 𝐵 = 𝐺 B=G 3. Perpendicular Bisector Points on a perpendicular bisector of a line segment are exactly equidistant from the segment’s endpoints. 4. Interior Angles of a Triangle The interior angles of a triangle add up to 18 0 ∘ 180 ∘ . 5. Isosceles Triangle Base angles are congruent. 6. Triangle Mid-Segment Theorem 𝐷 𝐸 DE is parallel to 𝐴 𝐶 AC. 𝐷 𝐸 DE is also half the length of 𝐴 𝐶 AC. 7. Centroid The centroid is where the medians of a triangle meet. The vertex to the centroid is 2 3 3 2 the length of the whole segment. 8. Parallelograms Properties: Opposite sides are parallel. Opposite angles are congruent. Opposite sides are congruent. Diagonals bisect each other. 9. Rectangles Properties: A rectangle is a type of parallelogram and it holds all the properties of a parallelogram but adds: 4 right angles. Diagonals are congruent. Questions Which of the following conditions or set of conditions must be met for a parallelogram to be a rectangle? Answer: (B) Diagonals are congruent. For what value of x is 𝑋 𝑌 𝑍 XYZ a rectangle? Answer: (C) 𝑥 = 4 x=4 What is the value of z for the rectangle? Answer: 𝑧 = 16 z=16 What is the value of x and y for the parallelogram? 𝑥 = 7 x=7 𝑦 = 3 y=3 In △ 𝑋 𝑌 𝑍 △XYZ, A is the centroid. 𝑍 𝐴 ‾ = 12 ZA =12, find 𝑍 𝐴 ‾ ZA and 𝐴 𝐷 ‾ AD . 𝑍 𝐴 ‾ = 8 ZA =8, 𝐴 𝐷 ‾ = 4 AD =4. Find the value of x. 𝑥 = 17 x=17 Find the value of x. 𝑥 = 4 x=4 What is the distance 𝑃 𝑄 PQ? 𝑥 = 156 𝑚 x=156m What is the length of 𝑄 𝑅 QR? 𝑛 = 3 n=3 Solve for x and y. 𝑥 = 8 0 ∘ x=80 ∘ 𝑦 = 4 0 ∘ y=40 ∘ Solve for x and y. 𝑥 = 11 3 ∘ x=113 ∘ 𝑦 = 6 7 ∘ y=67 ∘ 𝑧 = 4 9 ∘ z=49 ∘ What is ∠ 𝑁 𝑇 𝑀 ∠NTM? Answer: (B) 2 5 ∘ 25 ∘ What is the length of 𝐵 𝐻 BH? Answer: (C) 5 Find x, y, and z. 𝑥 = 11 3 ∘ x=113 ∘ 𝑦 = 6 7 ∘ y=67 ∘ 𝑧 = 4 9 ∘ z=49 ∘ Find x, y, and z. 𝑥 = 6 7 ∘ x=67 ∘ 𝑦 = 6 3 ∘ y=63 ∘ 𝑧 = 3 2 ∘ z=32 ∘ Use the figure at the right. If 𝑙 ∥ 𝑚 l∥m, what is 𝑚 ∠ 𝑙 m∠l? Answer: (B) 5 8 ∘ 58 ∘ Using the figure at the right, prove ∠ 1 = ∠ 8 ∠1=∠8. 𝑟 ∥ 𝑠 r∥s ∠ 6 ≅ ∠ 8 ∠6≅∠8 (Vertical Angle Congruence) ∠ 1 ≅ ∠ 6 ∠1≅∠6 (Corresponding Angles are Congruent) ∠ 1 ≅ ∠ 8 ∠1≅∠8 (Transitive Property) Using the rectangle below, prove △ 𝐴 𝐵 𝐸 ≅ △ 𝐶 𝐷 𝐸 △ABE≅△CDE. 𝐴 𝐵 𝐶 𝐷 ABCD is a rectangle. 𝐴 𝐵 ≅ 𝐶 𝐷 AB≅CD (Definition of a Rectangle) 𝐷 𝐸 ≅ 𝐵 𝐸 DE≅BE (Diagonals of a Rectangle Bisect Each Other) △ 𝐴 𝐵 𝐸 ≅ △ 𝐶 𝐷 𝐸 △ABE≅△CDE (SSS)