Unit 4. Triangle Relationships. Oct 3 8:20 AM. Oct 3 8:21 AM. Oct 3 8:26 AM. Oct 3 8:28 AM. Oct 3 8:27 AM. Oct 3 8:27 AM

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1 Unit 4 Triangle Relationships Classifying Triangles triangle -a figure formed by three segments joining three noncollinear points Classification of triangles: by sides by angles Oct 3 8:20 AM Oct 3 8:21 AM Classification by sides Oct 3 8:28 AM Oct 3 8:26 AM You try: Classification by angles Oct 3 8:27 AM Oct 3 8:27 AM 1

2 You try: Oct 3 8:31 AM Oct 3 8:32 AM Parts of a triangle Oct 3 8:33 AM Oct 3 8:34 AM Ticket out the Door Class activity: triangle classification with white-boards Ticket out the Door Assignment: pages #11-22 all, even Oct 3 8:38 AM Oct 3 8:34 AM 2

3 Angle Measures of Triangles Triangle Sum Theorem -The sum of the measures of the angles of a triangle is 180 degrees. Oct 3 8:40 AM Oct 3 11:31 AM corollary to a theorem -a statement that can be proved easily using the theorem Corollary to the Triangle Sum Theorem -The acute angles of a right triangle are complementary. Given m<a=25 and m<b=95, find m<c. C x 95 B 25 A Oct 3 11:32 AM Oct 3 11:33 AM You try: Triangle ABC and triangle CAD are right triangles. Suppose m<adc=50. A B D C Oct 3 12:13 PM Oct 3 12:14 PM 3

4 Interior angles -the three angles inside a triangle Exterior angles -formed when the sides of a triangles are extended -the angles that are adjacent to the interior angles -only one exterior angle per vertex Exterior Angle Theorem -The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Oct 3 12:14 PM Oct 3 12:16 PM You try: Given m<d=45 and m<dfg=100, find m<e. D 6. Find m<5. D G F E 100 G F E 5 Oct 3 12:17 PM Oct 3 12:20 PM 1. Ticket out the Door Oct 3 12:34 PM Oct 3 12:29 PM 4

5 Isosceles and Equilateral Triangles Assignment: pages #6-14 even, 15-16, 18, 20, 23 Let's make an isosceles triangle!! What do we notice about the sides of the triangle? What do we notice about the angles of the triangle? Oct 3 12:32 PM Oct 4 8:42 AM Parts of an Isosceles Triangle Base Angles Theorem -If two sides of a triangle are congruent, then the angles opposite them are congruent. base angle base angle m<b = m<c Oct 5 7:57 AM Oct 5 7:59 AM Find the measure of <L. Find the measure of <F. M F L? 52 N G 65 H What is this a picture of? Where is it located? Oct 5 8:03 AM Oct 5 8:07 AM 5

6 Converse of the Base Angles Theorem -If two angles of a triangle are congruent, then the sides opposite them are congruent. Find the value of x. R 4x + 4 AB = AC S 32 T Oct 5 8:06 AM Oct 5 8:09 AM You try: Equilateral Theorem -If a triangle is equilateral, then it is equiangular. Equiangular Theorem -If a triangle is equiangular, then it is equilateral. Oct 5 8:11 AM Oct 5 8:11 AM Let's try these together! D E 6x + 3 7x - 1 F Oct 5 8:13 AM Oct 5 8:15 AM 6

7 Class Activity: white board practice The Pythagorean Theorem and the Distance Formula Class-work: pages #7, 8, 10, 12, 14, 18, 19, 20, 22, 24 Drag these names over to their corresponding parts on this triangle. leg leg hypotenuse right angle Oct 5 8:15 AM Oct 3 12:35 PM The Pythagorean Theorem -only in a RIGHT triangle! a 2 + b 2 = c 2 What is a Pythagorean Triple? Find the length of the hypotenuse Why does this work? Let's watch this short video to find out. c Oct 4 8:04 AM Oct 4 8:06 AM You try: Find the unknown side length b Oct 4 8:08 AM Oct 4 8:09 AM 7

8 The Distance Formula -can be very useful when the coordinates are not already drawn in a plane Find the distance between the points J( 1, 7) and K(5, 1). Where did this crazy formula come from? Oct 4 8:10 AM Oct 4 8:11 AM Find the distance using the Pythagorean Theorem. Find the distance using the Distance Formula. Find the distance between A(0,0) and B(3,4). Oct 4 8:14 AM Oct 4 8:15 AM Use the Distance Formula to decide whether PQ=QR. Assignment: pages #8 30 even P(4, -4) Q(1, -6) R(-1, -3) Oct 4 8:26 AM Oct 4 8:17 AM 8

9 Find the distance between the points. 3. ( 2,3) and (3,0) 4. (7, 3) and ( 2, 1) Oct 15 2:05 PM Oct 16 10:52 AM The Converse of the Pythagorean Theorem Is triangle ABC a right triangle? C The Converse of the Pythagorean Theorem -If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle A IN OTHER WORDS... B 20 If a 2 + b 2 = c 2 works, then it is a right triangle. Oct 4 8:28 AM Oct 5 3:39 PM Is triangle DEF a right triangle? 4 D 3 Classifying Triangles Acute Note: BE CAREFUL here! Pay attention to the direction of your statement with the direction of your inequality! F E 6 Right Obtuse Oct 5 3:40 PM Oct 5 3:42 PM 9

10 Show that the triangle is an acute triangle. Show that the triangle is an obtuse triangle Oct 5 3:46 PM Oct 5 3:48 PM Classify the triangle as acute, right, or obtuse. Classify the triangle with the given side lengths as acute right, or obtuse. 4, 6, 7 12, 35, 37 9, 12, 13 16, 30, Oct 5 3:49 PM Oct 5 3:51 PM Bellwork 1. Find the distance between (3 8) and (3, 2). You try: 2. Find the distance between ( 1,3) and ( 6,5). 3. Find c of a right triangle if a= 6, and b= Find "a" of a rt. triangle if c = 17, and b=5. 5. Is the triangle a right triangle if the sides measure 11, 5, and 7? Mar 20 10:51 AM Oct 5 3:52 PM 10

11 Ticket out the Door Assignment: pages #10, 11, 12, 13, 16, 17, 18, 20, 22, 26, 28, 30, 36 Oct 5 3:53 PM Oct 5 3:54 PM Medians of a Triangle Oct 5 3:55 PM Oct 5 4:10 PM 4.6 median of a triangle -a segment from a vertex to the midpoint of the opposite side In triangle XYZ, draw a median from X to its opposite side. Be sure to label appropriately. Y X Z Re label with new lengths. Oct 5 4:12 PM Oct 5 4:15 PM 11

12 You try: Facts about medians: 1. The three medians in a triangle meet at a point inside the centroid triangle called the. center 2. The centroid is the point in a triangle. 3. The centroid divides a median into two pieces, a shorter piece and a longer piece. The longer piece is 2 times as long as the shorter piece. Oct 5 4:15 PM Oct 5 4:18 PM Point E is the centroid of triangle ABC and BF = 24. Find EF and BE. Oct 5 4:24 PM Oct 5 4:25 PM Oct 5 4:26 PM Oct 5 4:27 PM 12

13 You try: Ticket out the door Oct 5 4:26 PM Oct 5 4:28 PM Assignment: page 210 # Triangle Inequalities Oct 5 4:30 PM Oct 5 4:31 PM Theorems -If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. -If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Name the angles from largest to smallest. 9 A 11 B C 13 Oct 5 4:31 PM Oct 5 4:33 PM 13

14 Use triangle ABC below to list the sides in order from longest to shortest. Name the sides from longest to shortest. R 62 S T Oct 5 4:35 PM Oct 5 4:40 PM You try: Note: Not every group of three segments can be used to make a triangle! The lengths have to have a certain relationship! Triangle Inequality Activity Oct 5 4:36 PM Oct 5 4:37 PM Triangle Inequality -The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A d. 6, 7, 15 e. 6, 8, 14 f. 6, 9, 13 B C Which one would be the most important to check? Oct 5 4:41 PM Oct 5 4:42 PM 14

15 You try: Ticket out the Door Oct 5 4:44 PM Oct 5 4:44 PM Assignment: pages #12-22 even, even, 34, 36 Oct 5 4:45 PM 15