Formal Geometry - Chapter 5 Notes Name: 5.1 Identify and use perpendicular bisectors and angle bisectors in triangles. - Sketch a perpendicular bisector to segment AB - Put point C anywhere on the perpendicular bisector. - Connect C to endpoints of segment AB. - Conclusions? A B W Backwards: - W is equidistant from endpoints X and Y. - Conclusions? X Y Bisector Theorems If a point is on the bisector of a segment, then it is equidistant from the endpoints of that segment. If a point is equidistant from the endpoints of a segment, then it is on the bisector of the segment. Ex1 -Which segment is the bisector? -Find AB Ex2 -Which segment is the bisector? - Find WY Ex3 - Which segment is the bisector? -Solve for x. 1
Definitions Concurrent lines: Point of concurrency: Circumcenter: Circumcenter Theorem The bisectors of a intersect at a point called the circumcenter that is equidistant from the vertices of the. *This is true of all triangles, even though the example looks equilateral. Ex4. Where is the circumcenter of each type of triangle? Sketch it! a) Acute Triangle b) Obtuse Triangle c) Right Triangle Angle Bisector: Theorems for Angle Bisectors If a point is on the bisector of an angle, then it is equidistant from the sides of the angle (at a ). If a point in the interior of angle is equidistant from the sides of the angle, then it is on the angle bisector. 2
Ex5 a. Solve for x b. Solve for x Incenter: Point of Concurrency for an in a triangle. Theorem: The angle bisectors of a triangle intersect at the that is equidistant from each side of the triangle (at a ). *Incenter is always the triangle! Ex6 Solve for a, b, c. x is the incenter. CHALLENGE (+1 on HW) Factor: 12x 2 5x - 2 5.2 Notes Identify and use altitudes and median in triangles. Median: Line segment from the of a side to the opposite. Medians intersect at the. A, B, C are midpoints. Construct the Centroid D. PB = 15, PD = 10, DB = 5 GC = 9, GD = 6, DC = 3 QA = 12, QD = 8, DA = 4 See any patterns??? Centroid: Point of concurrency for medians. Theorem: The centroid is the distance from each to the of the opposite side. Write equations for the centroid based on PGQ: 3
Ex1 Given: Q is the centroid, BE = 9, Ex2 Find the centroid on a coordinate plane: find BQ = and QE = ABC: A(1,10), B(5,0), C(9,5), find the centroid. Where will the centroid be for each type of triangle? (Inside, Outside, or on the triangle?) Acute: Obtuse: Right: Altitude: Measures height from to opposite side at a. Altitudes intersect at. Orthocenter can be inside, outside, or on the triangle. Ex3 Find the Orthocenter: FGH: F(-2,4), G(4,4), H(1,-2). Summary: All Of My Children Are Bringing In Peanut Butter Cookies 4
5.3 Inequalities in One Triangle Inequalities Properties Properties apply to 1. a<b, 2. a = b, or 3. a>b. Transitive Property: Addition/Subtraction Properties: Ex1 If m<2 = 30 and m<3 = 40, find m<1 = If m<2 = 50 and m<3 = 40, find m<1 = If m<2 = 60 and m<3 = 40, find m<1 = Any patterns? Exterior Angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding angles. Exterior Angle Inequality: We know m<1 = m<2 + m<3, so <1 <3 and <1 <2. Ex2 List all angles less than m<7. Ex3 What is the longest side of ABC? Ex4 List the angles from smallest to largest. Which angle is it opposite? 5
Theorems -If one side of a triangle is longer than another side, then the angle the longer side has a measure than the shorter side. -If one angle of a triangle has a greater measure than another angle, then the side opposite the angle is than the side opposite the lesser angle. Ex5 List all side lengths from shortest to longest for the specified triangle. a) ABC b) BCD c) All (final answer) Ex6 List all angles (1, 2, 3, 4, 5, 6) from smallest to largest, given m<2 = m<5 Challenge! +1 on HW! Get y in terms of x. Hint: Factor out y! 3xy + 4y = 9 6
5.4 Indirect Proofs Indirect Reasoning: Proving something false Indirect Proof/Proof by Contradiction: Steps to write an indirect proof: Ex1 Negate the given statement a. <ABC <XYZ b. 2 is a factor of n c. <3 is an obtuse angle d. XY AB e. x > 5 Ex2 Algebraic Indirect Proof Given: -3x + 4 > 16 Prove: x < -4 Ex3 Write an indirect proof: If 7x > 56, then x > 8 7
Ex4 Geometric Indirect Proof Given: XZ > YZ Prove: <X <Y Ex5 #1 from Indirect Proof WS (Complete it on your worksheet!) *You need a ruler today! 5.5 Use the Triangle Inequality Theorem to identify possible triangles Ex1 Draw triangles with the following side lengths (as best as possible) a) 3cm., 4cm., 5cm. c) 2cm., 3cm., 4cm. b) 3cm., 3cm., 3cm. d) 2cm., 3cm., 5cm. e) 2cm., 3cm., 6cm. 8
What do you notice about the triangles drawn in Ex1? Can you come up with a rule to describe the possible side lengths of a triangle? Triangle Inequality Theorem: The sum of the lengths of any 2 of a triangle must be than the length of the third side. Ex: Ex2 Are these possible side lengths of a triangle? EXPLAIN. a) 8, 15, 17 b) 6, 8, 14 c) 2, 2, 2 d) 1, 2, 4 Ex3 Make a list for the sides of a triangle: a) That ARE possible lengths b) That are NOT possible lengths Ex4 Find the range of values for x. Ex5 Given: CA = CT, Prove: CL + LT > CA 9
Ex6 Given: GL = LK, Prove: JH + GH > JK 5.6 Apply the Hinge Theorem and its converse to make comparisons in two triangles Ex1 Look at 3 possible positions of a car jack. Notice the sides of the jack remain the same.how are they different? Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first triangle is than the included angle of the second triangle, then the third side of the first triangle is than the third side of the second triangle. Converse If two sides of a triangle are congruent to two sides of another triangle, and the third side in the first triangle is longer than the third side of the second triangle, then the included angle measure of the first triangle is greater than the included angle measure of the second triangle. 10
Ex2 Compare the measures a) WX and XY b) m<fcd and m<bfc Ex3 Find the range of possible values of x: a) b) Ex4 Given: AB AD, Prove: EB > ED 11
Ex5 Given: ST = PQ, SR = QR, SP > ST, Prove: m<srp > m<prq Ex6 Fill in the blank to write an inequality. Given: ZU UX. a) <ZUY <YUX b) WU YU c) <WUX <ZUW 12