5.1: Date: Geometry. A midsegment of a triangle is a connecting the of two sides of the triangle.

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1 5.1: Date: Geometry A midsegment of a triangle is a connecting the of two sides of the triangle. Theorem 5-1: Triangle Midsegment Theorem A If a segment joins the midpoints of two sides of a triangle, then the segment is to the third side and is as long. B C Ex 1). What are three pairs of parallel segments in DEF? Ex 2). In RST, G, H, and K are midpoints. If RS = 28, GH = 20, and RH = 22, what is the length of KH, ST, and GK? Ex 3). What is the distance across the river?

2 Ex 4). Ex 5). Homework: pg. 305 #1 4, 7 22

3 5.2: Date: Geometry Midsegments of Triangles Refresher (5.1): Ex 1). Name all of the parallel sides: Ex 2). Find the value of x: Theorem 5-2: Perpendicular Bisector Theorem if then If a point is on the perpendicular bisector of a segment, then it is from the endpoints of the segment. Theorem 5-3: Converse of 5-2 if then If a point is equidistant from the endpoints of a segment, then it is on the perpendicular of the segment. Ex 3). What is MN? Ex 4). The monkey bars are located midway between

4 the slide and the tether ball. Which of the following are about equidistant from the slide and the tether ball? Theorem 5-4: Angle Bisector Theorem if then If a point is on the bisector of an angle, then the point is from the sides of the angle. Theorem 5-5: Converse of 5-4 if then If a point in the of an angle is equidistant from the sides of the angle, then the point is on the angle bisector. Ex 5). What is OP? Ex 6). What is m HEG? Homework: pg. 312 #1 3, 6 22

5 5.3 Day 1: Date: Geometry When three or more lines intersect at point, they are. The point at which they intersect is the of. Theorem 5-6: Concurrency of Perpendicular Bisectors Theorem The point of concurrency of the perpendicular bisectors of a triangle is called the of the triangle. The circumcenter is the of the circle. *The circumcenter of a triangle can be inside, on, or outside a triangle* Ex 1). Find the circumcenter of the triangle with vertices A(0, 4), B(0, 0), and C(8, 0).

6 Ex 2). Find the circumcenter of the triangle with vertices A(2, 7), B(10, 7), and C(10,3). Ex 3). A civil engineer wants to install a cell phone tower that is equidistant from the mall, the airport, and the subway. How should the engineer determine where to build the tower? Mall Airport Subway Homework: pg. 319 # 4 10

7 5.3 Day 2: Date: Geometry The point of concurrency of the bisectors of a triangle is called the of the triangle. For any triangle, the incenter is always the triangle. Theorem 5-7: Concurrency of Angle Bisectors Theorem In the diagram, X, Y, and Z are equidistant from P, the incenter. P is the center of the circle that is in the triangle. (See diagram on pg. 320) Ex 1). GB = 8x 7 and GD = 5x + 8. What is GF?

8 Ex 2). What is the value of x? Ex 3). What is the value of x? Homework: pg. 321 #1 10, 12 17

9 5.4 Day1: Date: Geometry A of a triangle is a segment whose endpoints are a and the of the opposite side. A triangle s medians are always. The point of concurrency of the medians is called the of the triangle. o This point is also called the of of a triangle. o The centroid is always the triangle. Theorem 5-8: Concurrency of Medians Theorem The medians are concurrent at a point that is the distance from each vertex to the midpoint of the opposite side. Ex 1). The centroid is shown. Find the lengths. a) Given: BF = 24. Find EF and BE. b) Given: BD = 9. Find BE and ED.

10 c) Given: KH = 12. Find JH. d) Given JK = 4. Find JG and KG. An of a triangle is the perpendicular segment from the vertex of the triangle to the line containing the opposite side. An altitude can be,, or the of a triangle. Ex 2). a) Is AC a median, an altitude, or neither? Explain. b) Is AE a median, an altitude, or neither? Explain. Homework: pg. 327 #1 17

11 5.4 Day 2: Date: Geometry The lines that contain the of a triangle are at the of the triangle. The orthocenter can be,, or the triangle. Ex 1). Find the orthocenter. What are the coordinates of the orthocenter of DEF?

12 Ex 2). What are the coordinates of the orthocenter of DEF if D(-2, -1), E(1, 6), and F(5, - 1)? Homework: pg. 331 #1 8, (use provided grids for #1, 4, 5, 6)

13 5.6 Day 1: Date: Geometry Comparison Property of Inequality: If a = b + c and c > 0, then. Corollary to the Triangle Exterior Angle Theorem: The measure of an exterior angle of a triangle is than the measure of each of its remote interior angles. Ex 1). Explain why m 1 > m 4. Theorem 5-10: If two sides of a triangle are not congruent, then the angle lies opposite the side. Ex 2). Which corner of the triangular plot of land forms the largest angle?

14 Theorem 5-11: If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. Ex 3). Order the sides of LMN in order from shortest to longest. Ex 4). Name the angles of ABC from largest to smallest. Homework: pg. 346 #4 22

15 5.6 Day 2: Date: Geometry Not every group of three segments can be used to form a triangle. The lengths of the segments must have the following relationship: Theorem 5-12: Triangle Inequality Theorem The of the lengths of any two sides of a triangle is than the length of the third side. Ex 1). Can a triangle have sides with the given lengths? Explain. a) 6, 8, 13 b) 3, 9, 12 Ex 2). Two sides of a triangle are 7 inches and 9 inches. What is the range of possible lengths for the third side? Ex 3). Two sides of a triangle are 2 inches and 12 inches. What is the range of possible lengths for the third side? Homework: pg. 350 #1, 3 12

16 5.7: Date: Geometry Theorem 5-13: The Hinge Theorem (SAS Inequality Theorem) If two sides of one triangle are congruent to two sides of another triangle, and the included angles are congruent, then the third side is opposite the included angle. Ex 1). Which of the following statements must be true? [A] AC = ST [B] RS > AC [C] AC < RT [D] AC > RT Ex 2). Which cheerleader s hands are futher apart? Theorem 5-14: Converse of the Hinge Theorem (SSS Inequality) If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger angle is opposite the longer third side.

17 Ex 3). What is the range of possible values for x? Ex 4). What is the range of possible values for x? Ex 5). What is the range of possible values for x? Homework: pg. 356 #1, 2, 6 12, 14 16

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