Incredibly, in any triangle the three lines for any of the following are concurrent.

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1 Name: Day 8: Circumcenter and Incenter Date: Geometry CC Module 1 A Opening Exercise: a) Identify the construction that matches each diagram. Diagram 1 Diagram 2 Diagram 3 Diagram 4 A C D A C B A C B C' B D A' B' Term Definition Diagram Point of Concurrency Where three or more lines in one place. Circumcenter The intersection of the of any triangle. *Center of the circumcircle of the triangle* Incenter The intersection of the of any triangle. *Center of the incircle of the triangle* Incredibly, in any triangle the three lines for any of the following are concurrent. Perpendicular Bisectors(circumcenter) Angle Bisectors(incenter) Medians( ) Altitudes( ) 1

2 Example 1: a) Construct the perpendicular bisectors of the three sides of the triangle below. b) This point of concurrency is called the. Example 2: a) Use the triangle below to construct the angle bisectors of each angle in the triangle. A B C b) This point of concurrency is called the. 2

3 Example 3: Use the diagram to identify where each point of concurrency will lie. a) The incenter will lie on (1) AD (2) AE (3) AF (4) GF b) The circumcenter will lie on (1) AD (2) AE (3) AF (4) GF Practice NYTS (Now You Try Some) 1. The diagram below shows the construction of the center of the circle circumscribed about. This construction represents how to find the intersection of 1) the angle bisectors of 2) the medians to the sides of 3) the altitudes to the sides of 4) the perpendicular bisectors of the sides of 2. Which geometric principle is used in the construction shown below? 1) The intersection of the angle bisectors of a triangle is the center of the inscribed circle 2) The intersection of the angle bisectors of a triangle is the center of the circumscribed circle 3) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the inscribed circle. 4) The intersection of the perpendicular bisectors of the sides of a triangle is the center of the circumscribed circle 3

4 Name: Day 9: Pythagorean Theorem Date: Geometry CC Module 1 A Opening Exercise: CD is the perpendicular bisector of AB at M. Which pair of segments does not have to be congruent in the construction shown? 1) AM, BM 2) AC, BC 3) CM, DM 4) AD, BD Term Definition Diagram Pythagorean Theorem a 2 b 2 c 2 The lengths of any triangle satisfy the Pythagorean theorem. Common Pythagorean Triples 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 (multiples of these numbers also satisfy the Pythagorean theorem.) Example 1: A 10 foot ladder leans against a building, as shown in the diagram below. If the bottom of the ladder is placed 6 feet from the base of the wall, how many feet up the wall does the brace reach? 4

5 Example 2: Tanya runs diagonally across a rectangular field that has a length of 40 yards and a width of 30 yards, as shown in the diagram below. What is the length of the diagonal, in yards, that Tanya runs? 1) 50 2) 60 3) 70 4) 80 Example 3: Which set of numbers could be the lengths of the sides of a right triangle? 1) 2) 3) 4) Example 4: Cole placed a ladder against the side of his house. How many feet from the base of a house must a 39-foot ladder be placed so that the top of the ladder will reach a point on the house 36 feet from the ground? Example 5: A cable 20 feet long connects the top of a flagpole to a point on the ground that is 16 feet from the base of the pole. How tall is the flagpole? 1) 8 ft 2) 10 ft 3) 12 ft 4) 26 ft Example 6: Don placed a ladder against the side of his house as shown in the diagram below. To the nearest foot what is the distance, x, from the foot of the ladder to the base of the house? 5

6 Practice NYTS(Now You Try Some!) 1. If the length of a rectangular television screen is 20 inches and its height is 15 inches, what is the length of its diagonal, in inches? 1) 15 2) ) 25 4) An 18-foot ladder leans against the wall of a building. The base of the ladder is 9 feet from the building on level ground. How many feet up the wall, to the nearest tenth of a foot, is the top of the ladder? 3. Which set of numbers does not represent the sides of a right triangle? 1) 2) 3) 4) 4. A 10-foot ladder is placed against the side of a building as shown in figure 1 below. The bottom of the ladder is 8 feet from the base of the building. In order to increase the reach of the ladder against the building, it is moved 4 feet closer to the base of the building as shown in figure 2. To the nearest foot, how much further up the building does the ladder now reach? Show how you arrived at your answer. 6

7 Name: LDay8and9: Pythagorean Theorem WHO DUNNIT Date: Geometry CC Module 1 A 7

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9 Name: Day 10: Mid-Segments in Triangles Date: Geometry CC Module 1 A Opening Exercise: 1. Campsite A and campsite B are located directly opposite each other on the shores of Lake Omega, as shown in the diagram below. The two campsites form a right triangle with Sam s position, S. The distance from campsite B to Sam s position is 1,300 yards, and campsite A is 1,700 yards from his position. What is the distance from campsite A to campsite B, to the nearest yard? 1) 1,095 2) 1,096 3) 2,140 4) 2, Find the midpoints of sides DE and FE in the triangle below using the perpendicular bisector construction. Label the points M and N then connect them to create mid-segment MN Term Definition Diagram B Mid-Segments in a The segment joining two of two sides of a triangle will always be : to the third side. A D E C the length of the third side. DE AC 1 DE AC 2 9

10 Example 1: In the diagram below, joins the midpoints of two sides of. Which statement is not true? 1) 2) 3) 2DC AC 4) 2AB DE Example 2: Determine the missing information. A. y 8 cm x B. x 10 cm y 7 cm C. 7 cm y x 12.5 cm 22 cm x = cm y = cm x = cm y = cm x = cm y = cm D. E. F. x 2x + 3 cm 34 cm 2x + 1 cm 5x - 4 cm 8 cm 8.5 cm y x = cm x = cm x = cm y = cm Example 3: In the diagram below of, is a midsegment of,,, and. Find the perimeter of. 10

11 Example 4: Determine the missing information. 18 cm 5 cm x 6 cm 5 cm y x 19 cm y 4 cm 6 cm 14 cm y 7 cm x x = cm y = cm x = cm y = cm x = cm y = cm Example 5: In the diagram of shown below, D is the midpoint of, E is the midpoint of, and F is the midpoint of. If,, and, what is the perimeter of trapezoid ABEF? 1) 24 2) 36 3) 40 4) 44 Practice NYTS (Now You Try Some!) cm x y 8.5 cm x = cm y = cm x = cm 11

12 3. 4. y 13 cm 5x + 1 cm 6 cm 16 cm x x = cm x = cm y = cm 1) 35 2) 32 3) 24 4) As shown in the diagram below, M, R, and T are midpoints of the sides of. If,, and, what is the perimeter of quadrilateral ACRM? 6. In the diagram below of, and are midsegments. If, and, determine and state the perimeter of quadrilateral FDEC. 7. Identify the construction that matches each diagram. A Diagram 1 Diagram 2 Diagram 3 Diagram 4 C D C C B A C' B A B D A' B' 12

Name: Day 11: Angle Sum of Triangles/Isosceles Triangles Date: Geometry CC Module 1 A Opening Exercise: In the diagram of shown below, D is the midpoint of, E is the midpoint of, and F is the

13 Name: Day 11: Angle Sum of Triangles/Isosceles Triangles Date: Geometry CC Module 1 A Opening Exercise: In the diagram of shown below, D is the midpoint of, E is the midpoint of, and F is the midpoint of. If,, and, what is the perimeter of parallelogram ADEF? Fill in the Fact/Discovery column based on geometry facts you have learned! Fact Diagram Ð Sum of D The 3 angles of any triangle sum to. Isosceles D A triangle with 2 congruent and 2 congruent. Straight Angle An angle that measures exactly. Side Lengths in a Triangle LONGEST SIDE of a triangle is always opposite the ANGLE. Angle Measures in a Triangle SHORTEST SIDE of a triangle is always opposite the ANGLE. 13

14 Example 1: Determine the measure of the missing angles in each triangle below then name the longest side of the triangle. a) m A = b) m C = c) m C = B 99 B 118 B 48 A 41 C A C A C b) Longest side: b) Longest side: b) Longest side: Example 2: In the diagram below of isosceles triangle ABC, and angle bisectors,, and are drawn and intersect at X. If m BAC 60 o, find. Example 3: Determine the measure of the numbered angles and EXPLAIN at least one fact you know about triangles that helped you reach your answer. m 1 = m 2 =

15 Example 4: In the diagram below of, H is a point on,,, and. Determine whether is an isosceles triangle and EXPLAIN at least two facts you know about triangles that helped you reach your answer. Practice NYTS (Now You Try Some!) 1.The accompanying diagram shows the roof of a house that is in the shape of an isosceles triangle. The vertex angle formed at the peak of the roof is 84. What is the measure of x? 1) 138 2) 96 3) 84 4) Determine the measure of the missing angles in each triangle below then name the longest side of the triangle. a) x = b) m C = c) m C = b) Longest side: b) Longest side: b) Longest side: 3. In the accompanying diagram of, is an equilateral triangle and. What is the value of x, in degrees? Explain at least 2 facts you know about triangles to show how you reached your solution. 15

16 Name: Day 10&11LabLesson: Triangle Inequality Opening Exercise: Date: Geometry CC Module 1 A For each statement, fill in the circle with >, < or =. For #4-5 use the diagram above TASK #1 TRIANGLE INVESTIGATION ACTIVITY!! During this activity, you will compare the sum of the measures of any two sides of a triangle with the measure of the third side You will pick three sides from your bag that create a triangle, and record the measures of each side of the triangle from shortest to longest; then, find the sum of the measures of the short and medium sides. Repeat this activity twice, with two other triangles, to complete the chart. [The lengths of the exploragons are labeled along the side of each piece] 1 Short side Medium side Long side Small + medium 2 3 TASK #2 You will pick three sides from your bag so that it is impossible to create a triangle, and record the measures of each side of the non-triangle from shortest to longest; then, find the sum of the measures of the short and medium sides. Repeat this activity twice, with two other non- triangles, to complete the chart. [The lengths of the exploragons are labeled along the side of each piece] 1 Short side Medium side Long side Short + medium

17 TASK #3 A) Compare the sum of the measures of the small and medium sides to the measure of the large side for each SUCCESSFUL triangle you created. Fill in the blank to describe what you notice. TRIANGLE INEQUALITY THEOREM In order to side lengths to create a triangle the of the 2 smaller sides must be than the longest side. Practice on Socrative! Socrative.com or Socrative the app Student Login - Room Name: WISEYMEPHAM 17

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19 Name: Day 12: Exterior Angle Theorem Date: Geometry CC Module 1 A Opening Exercise: In the diagram below, is isosceles with. If and, what is? Explain at least 2 facts you know about triangles to show how you reached your solution. Discovery! In the diagram of below,,, and is extended through N. Determine the measure of the angles below. m LKM m LKN What do you notice about the measure of LKN compared to the measures of L and M? Fact Diagram Ð Exterior of a Theorem D The measure of an exterior angle of a triangle is equal to the of the measures of the two interior angles of the triangle. m ACD m A m B 19

20 Example 1: Find m 2. Example 2: Find m 5, m 6 and m 7. Example 3: Given with m B 56 o and side extended to D, as shown below. Which value of x makes? 1) 59º 2) 62º 3) 118º 4) 121º Example 4: In the diagram below of with side extended through D, m A 50 o and m BCD 120 o. Find the missing angles of and determine which side of is the longest side? Justify your answer. Example 5: In the diagram of below, is extended to point D. If,,, what is? 20

21 Practice NYTS (Now You Try Some!) 1. Determine the m Determine the m Solve for x in the diagram below. 4. Find the value of x a) x = b) x = x x 5. In the diagram below of isosceles, the measure of vertex angle B is 80. If extends to point D, find m BCD? 21

22 Name: LabLessonDay12and 13: Angle Sum of a Triangle Date: Geometry CC Module 1 A 22

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24 Name: Day 13: Module 1 Constructions and Triangle Basics Test Review Date: Geometry CC Determine whether the following are (T)rue or (F)alse. 1. The incenter is the point of concurrency of the perpendicular bisectors of a triangle. T or F 2. A triangle with side lengths 7, 8, 15 will successfully make a triangle. T or F 3. An median in a triangle is a segment from one vertex and perpendicular to the opposite side. T or F 4. A triangle with side lengths 7, 24, 26 successfully make a right triangle. T or F 5. In the diagram below, joins the midpoints of two sides of. Which statement is not true? 1) 2) 3) 2DC AC 4) 2AB DE 6. If the measures of the angles of a triangle are represented by,, and, the triangle is 1) an isosceles triangle 2) a right triangle 3) an acute triangle 4) an equiangular triangle 7. In the diagram below of isosceles, the measure of vertex angle B is 80. If extends to point D, what is? 1) 50 2) 80 3) 100 4)

25 8. The end of a dog's leash is attached to the top of a 5-foot-tall fence post, as shown in the diagram below. The dog is 7 feet away from the base of the fence post. How long is the leash, to the nearest tenth of a foot? 1) 4.9 2) 8.6 3) 9.0 4) CD is the perpendicular bisector of AB at M. Which pair of segments does not have to be congruent in the construction shown? 1) AM, BM 2) AC, BC 3) CM, DM 4) AD, BD 10. In the diagram below of with side extended, m A 42 o and the exterior angle at C measures 112 o. Which side of is the longest side? Justify your answer. 11. In shown below, L is the midpoint of, M is the midpoint of, and N is the midpoint of. If,, and, the perimeter of parallelogram BMNL is 25

26 12. A woman has a ladder that is 26 feet long. If she sets the base of the ladder on level ground 10 feet from the side of a house, how many feet above the ground will the top of the ladder be when it rests against the house? 13. In the diagram below of isosceles triangle MNO, MN ON and and are bisected by and, respectively. Segments MS and intersect at T, and. If, then the measure of angle OTM is Note: Not drawn to scale 14. Construct an equilateral triangle using the segment shown as one of the three equal sides. Leave all construction marks. D C 26

27 15. Bisect the angle below. A 16. Using a compass and a straightedge, construct the circumcenter of ABC. Label it O. 17. Using a compass and a straightedge, construct the incenter of ABC. Label it O. 27

28 18. In the diagram of below, is extended to point D. If,,, what is? 19. Find the measure of the missing numbered angles. Explain at least 2 facts you know about triangles to show how you reached your solutions. m 1 = m 2 = m 3 = Using a compass and straightedge, construct a perpendicular line(altitude) from vertex J to. [Leave all construction marks.] 28

1. AREAS. Geometry 199. A. Rectangle = base altitude = bh. B. Parallelogram = base altitude = bh. C. Rhombus = 1 product of the diagonals = 1 dd

1. AREAS. Geometry 199. A. Rectangle = base altitude = bh. B. Parallelogram = base altitude = bh. C. Rhombus = 1 product of the diagonals = 1 dd Geometry 199 1. AREAS A. Rectangle = base altitude = bh Area = 40 B. Parallelogram = base altitude = bh Area = 40 Notice that the altitude is different from the side. It is always shorter than the second