Geometry Chapter 5 - Properties and Attributes of Triangles Segments in Triangles

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1 Geometry hapter 5 - roperties and ttributes of Triangles Segments in Triangles Lesson 1: erpendicular and ngle isectors equidistant Triangle congruence theorems can be used to prove theorems about equidistant points. Distance and erpendicular isectors Theorem Hypothesis onclusion erpendicular isector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. l X Y onverse of the erpendicular isector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment. l X Y Locus The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment. pplying the erpendicular isector Theorem and Its onverse Ex1: Find each measure. NM =. =. TU = N D M 12 Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line. 38 U 3x + 9 7x - 17 T

2 Distance and ngle isectors Theorem Hypothesis onclusion ngle isector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. onverse of the ngle isector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. pplying the ngle isector Theorems Ex2: Find each measure. =. m EFH, given that m EFG = 50. m MKL

3 Ex4: Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints (6, -5), and D(10, 1). Geometry Lesson 2: isectors of Triangles Since a triangle has three sides, it has three perpendicular bisectors. When you construct the perpendicular bisectors, you find that they have an interesting property. concurrent point of concurrency circumcenter of the triangle The circumcenter of a triangle is equidistant from the vertices of the triangle. ircumcenter Theorem The circumcenter can be inside the triangle, outside the triangle, or on the triangle. cute triangle Obtuse triangle Right triangle

4 The circumcenter of Δ is the center of its circumscribed circle. ircumscribed circle Ex1: DG, EG, and FG are the perpendicular bisectors of Δ. Find G. Using roperties of erpendicular isectors Ex2: Find the circumcenter of ΔHJK with vertices H(0, 0), J(10, 0), and K(0, 6). triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. incenter of a triangle Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle.

5 Unlike the circumcenter, the incenter is always inside the triangle. cute triangle Obtuse triangle Right triangle The incenter is the center of the triangle's inscribed circle. Inscribed circle Ex3: M and L are angle bisectors of ΔLMN. Find each measure.. the distance from to. MN. m MN Using roperties of ngle isectors Ex4: city planner wants to build a new library between a school, a post office, and a hospital. Draw a sketch to show where the library should be placed so it is the same distance from all three buildings. S L Geometry Lesson 3: Medians and ltitudes of Triangles median of a triangle D Every triangle has three medians, and the medians are concurrent. centroid of the triangle The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangluar region will balance.

6 The centroid of a triangle is located opposite side. 2 3 entroid Theorem of the distance from each vertex to the midpoint of the *Remember, the centroid is closer to each side than to the verte Using the entroid to Find Segment Lengths Ex1: In ΔLMN, RL = 21, and SQ = 4. Find. LS =. NQ = Ex2: sculptor is shaping a triangular piece of iron that will balance on the point of a cone. t what coordinates will the triangular region balance?

7 altitude of a triangle Every triangle has three altitudes. n altitude can be inside, outside, or on the triangle. orthocenter of a triangle Geometry Lesson 4: The Triangle Midsegment Theorem Q midsegment of a triangle R midsegments: midsegment triangle: Every triangle has three midsegments, which form the midsegment triangle.

8 Examining Midsegments in the oordinate lane Ex1: The vertices of ΔXYZ are X(-1, 8), Y(9, 2), and Z(3, -4). M and N are the midpoints of XZ YZ. Show that. MN // XY. MN = 1 2 XY. The relationship shown in Example 1 is true for the midsegment of every triangle. Triangle Midsegment Theorem midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.

9 Ex2: Find each measure.. D Using the Triangle Midsegment Theorem Ex3: In an -frame support, the distance Q is 46 inches. What is the length of the support ST if S and T are at the midpoints of the sides?. m D Geometry Relationships in Triangles Lesson 5: Indirect roof and Inequalities in One Triangle You have written proofs using direct reasoning. That is, you began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction. 1. Identify the conjecture to be proven. Writing an Indirect roof 2. ssume the opposite (the negation) of the conclusion is true. 3. Use direct reasoning to show that the assumption leads to a contradiction. 4. onclude that since the assumption is false, the original conjecture must be true. Writing an Indirect roof Ex1: Write an indirect proof that a right triangle cannot have an obtuse angle.

10 Ex1:Write an indirect proof that if a > 0, then 1 a > 0. ngle-side Relationships in Triangles Theorem Hypothesis onclusion If two sides of a triangle are not congruent, then the larger angle is opposite the longer side If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. Y Z X Ordering Triangle Side Lengths and ngle Measures Ex2: Write the angles in order from smallest to largest.. Write the sides in order from shortest to longest. triangle is formed by three segments, but not every set of three segments can form a triangle. The sum of any two side lengths of a triangle is greater than the third length. Triangle Inequality

11 pplying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. Ex1: 3, 5, 7. 4, 6.5, 11. n + 5, n 2, 2n, when n = 3 Finding Side Lengths Ex4: The lengths of two sides of a triangle are 8 in. and 13 in. Find the range of possible lengths for the third side. Ex5: The figure shows the approximate distances between cities in alifornia. What is the range of distances from San Francisco to Oakland? Geometry Lesson 6: Inequalities in Two Triangles Inequalities in Two Triangles Theorem Hypothesis onclusion Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. E D F

12 Ex1:ompare m and m D. Using the Hinge Theorem : ompare EF and FG. : Find the range of values for k. Ex2: John and Luke leave school at the same time. John rides his bike 3 blocks west and 4 blocks north. Luke rides 4 blocks east and then 3 blocks at a bearing of N 10 E. Who is farther from school? Ex3: Write a two-column proof. Given: D, m D > m D rove: D > roving Triangle Relationships Statement Reason