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Chapter 5 TEST Study Guide Scheduled for Thursday, March 9, 2017 - Lines (Lesson 1) - Angles of Triangles (Lesson 3) -Pythagorean Theorem(Lesson 5) -Use the Pythagorean Theorem (Lesson 6) -Distance on the Coordinate Plane (Lesson 7) Lesson 1 - Lines In the figure above, lines m and n are parallel to one another and are intersecting by a line called a “Transversal.” When this happens, special angles are made. Adjacent Angles-Angles that are NEXT TO each other. They share a common side. These angles are S UPPLEMENTARY which means the sum of these two angles is equal to 180 DEGREES. Adjacent angles: < 1 & < 2 <3 &<4 <5 &<6 <7 &<8 <1 &<3 <2 &<4 <5 &<7 <6 &<8 Vertical Angles- Angles that are OPPOSITE from each other. They share a common vertex. These angles are C ONGRUENT which means they are equal in measure. Vertical angles: < 1 & < 4 <2 &<3 <5 &<8 <6 &<7 Corresponding Angles-When two parallel lines are crossed by another line (which is called the Transversal), the angles in matching corners are called corresponding angles. These angles are C ONGRUENT. (“same spot different group”) Corresponding Angles: < 1 & < 5 <2 &<6 <3 &<7 <4 &<8 Alternate Interior Angles-When two parallel lines are crossed by another line (which is called the Transversal), the pairs of angles on opposite sides of the transversal but inside the two lines are called Alternate Interior Angles. These angles are CONGRUENT. (“inside the walls of the parallel lines”) Alternate Interior Angles: < 3 & < 6 <4 &<5 Alternate Exterior Angles-When two parallel lines are crossed by another line (which is called the Transversal), the pairs of angles on opposite sides of the transversal but outside the two lines are called Alternate Exterior Angles. These angles are CONGRUENT. (“outside the walls of the parallel lines”) Alternate Exterior Angles: < 1 & < 8 <2 &<7 Finding Missing Angles Suppose you know that m<1 = 50 ° You can use that information to figure out the measures of angles 2, 3, and 4. YOU MUST JUSTIFY USING VOCABULARY TERMS!! (vertical, supplementary, corresponding, alt. interior, alt. exterior) m < 2 = 130 because < 1 and < 2 are supplementary angles (add up to 180) m < 3 = 50 because < 1 and < 3 are vertical angles m < 4 = 130 < 1 and < 4 are supplementary or < 4 and < 2 are vertical angles. Lesson 3 – Angles of Triangles All INTERIOR angles in a triangles add to give you 180° Finding Interior Angles of Triangles Find the value of c in the given triangle 38 + 85 + c = 180 123 + c = 180 C = 57° The missing angle is 57°. Find the value of x in the given triangle 45°, 65° 1 → Add the two angles given 45 + 65 = 110 nd 2 → Subtract the sum from 180 (180° in total triangle) 180-110 =70 The missing angle is 70°. st Finding Exterior Angles of Triangles interior angle + interior angle = exterior angle The two “interior angles” are x° and 22°. The “exterior angle” is 134°. x + 22 = 134 (interior + interior = exterior) x = 112° The two “interior angles” are x° and 58°. The “exterior angle” is 120°. x + 58 = 120 (interior + interior = exterior) x = 62° Lesson 5 – Pythagorean Theorem The formula is a2 + b2 = c2 Pythagorean Theorem- In a right triangle, the square of the measure of the hypotenuse is equal to the sum of the measure of the squares of the legs. Example- “c” is missing 52 + 122 = c2 25 + 144 = c2 169 = c2 c2 = 169 c = √169 c = 13 Example-- “a” or “b” is missing Example: Is this triangle a right triangle? Prove using the Pythagorean Theorem Does a2 + b2 = c2 ? *The longest side is always the hypotenuse (c). ● ● ● ● a2 + b2 = c2 102 + 242 = 262 100 + 576 = 676 676 = 676 They are equal, so ... Yes, it is a Right Triangle! Example: Is a triangle with lengths 8 in,15 in.,and 16 in. a right triangle? Does 82 + 152 = 162 ? 82 + 152 = 162 ● ● 64 + 225 = 256, ● 289 256 They are NOT equal so, NO, it is not a right triangle! Example: Using the Pythagorean theorem twice -For triangle FBD: We know that leg FB= 6 cm. We have to find leg BD. Then hypotenuse FD Triangle: ABD- leg AD= 4 cm, leg AB= 3 cm. BD is hypotenuse of this triangle. Use the Pythagorean Theorem to find BD (the hypotenuse of triangle ABC) 42+32=c2 16 + 9 = c2 25 = c2 √25 = c 5=c (BD = 5 cm) Use the Pythagorean theorem to find the hypotenuse FD. (legs are BD & FB) 52+62=c2 25 + 36 = c2 61 = c2 √61 = c 7.8 = c Answer: FD = 7.8 cm Lesson 6 – Use the Pythagorean Theorem Example: The formula is a2 + b2 = c2 A 13 feet ladder is placed 5 feet away from a wall. The distance from the ground straight up to the top of the wall is 13 feet Will the ladder the top of the wall? Let the length of the ladder represents the length of the hypotenuse or c = 13 and a = 5 the distance from the ladder to the wall. c2 = a2 + b2 132 = 52 + b2 169 = 25 + b2 169 - 25 = 25 - 25 + b2 (minus 25 from both sides to isolate b2 ) 144 = 0 + b2 144 = b2 b = √144 = 12 The ladder will never reach the top since it will only reach 12 feet high from the ground yet the top is 14 feet high. Example: c2 = a2 + b2 c2 = 162 + 122 c2 = 256 + 144 c2 = 400 c = 20 ft Lesson 7–Distance on the Coordinate Plane You can use the Pythagorean Theorem to find the distance between two points on the coordinate plane. Steps: 1. Graph the two points 2. Draw a right triangle 3. Count the boxes to find the length of the two legs (a and b) 4. Find the hypotenuse using the Pythagorean Theorem Example: Graph the ordered pairs (2, –3) and (5, 4). Then find the distance e between the two points. Step 1: Graph the two points Step 2: Draw a right triangle Step 3: Count the boxes → a = 3, b = 7 Step 4: Use the pythagorean theorem a2 + b2 = c2 32 + 72 = c2 58 = c2 = 7.6 ≈ c The Pythagorean Theorem Replace a with 3 and b with 7. 2 3 + 72 = 9 + 49, or 58. Definition of square root Use a calculator. The points are about 7.6 units apart.