Relationships within Triangles Chapter 6 Perpendicular and Angle

Relationships within Triangles Chapter 6

Perpendicular and Angle Bisectors I can use perpendicular and angle bisectors to find measures and distance relationships.

Perpendicular and Angle Bisectors Vocabulary (page 166 in Student Journal) equidistant: a point that is the same distance away from 2 objects

Perpendicular and Angle Bisectors Core Concepts (pages 166 and 167 in Student Journal) Perpendicular Bisector Theorem In a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Perpendicular and Angle Bisectors Converse of the Perpendicular Bisector Theorem In a plane, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

Perpendicular and Angle Bisectors Angle Bisector Theorem If a point lies on the bisector of an angle, then it is equidistant from the 2 sides of the angle. Converse of the Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from 2 sides of the angle, then it lies on the bisector of the angle.

Perpendicular and Angle Bisectors Examples (page 168 in Student Journal) Find the measure. #3) SU

Perpendicular and Angle Bisectors Solution #3) 6

Perpendicular and Angle Bisectors Find the equation of the perpendicular bisector. #4)

Perpendicular and Angle Bisectors Solution #4) y – 0 = -1/2(x – 1)

Perpendicular and Angle Bisectors Find the measure. #7) BD

Perpendicular and Angle Bisectors Solution #7) 4

Bisectors in Triangles I can use and find the circumcenter and incenter of a triangle.

Bisectors in Triangles Vocabulary (page 171 in Student Journal) concurrent: 3 or more lines intersect at the same point of concurrency: the point where 3 or more lines intersect

Bisectors in Triangles circumcenter: the point of concurrency of the perpendicular bisectors of a triangle incenter: the point of concurrency of the angle bisectors of a triangle

Bisectors of Triangles Core Concepts (pages 171 and 172 in Student Journal) Circumcenter Theorem The perpendicular bisectors of the sides of a triangle are concurrent at a point (circumcenter) equidistant from the vertices.

Bisectors of Triangles Incenter Theorem The bisectors of the angles of a triangle are concurrent at a point (incenter) equidistant from the sides of the triangles.

Bisectors of Triangles Examples (pages 172 and 173 in Student Journal) Point N is the incenter of the triangle. #1) ND = 2 x – 5 NE = -2 x + 7 Find NF.

Bisectors of Triangles Solution #1) 1

Bisectors of Triangles Find the indicator measure. #5) PS

Bisectors of Triangles Solution #5) 8

Bisectors of Triangles Find the coordinates of the circumcenter of the triangle. #8) A(-2, -2), B(-2, 4), C(6, 4)

Bisectors of Triangles Solution #8) (2, 1)

Medians and Altitudes of Triangles I can use medians and find centroids of triangles and use altitudes and find the orthocenters of triangles.

Medians and Altitudes of Triangles Vocabulary (page 176 in Student Journal) median of a triangle: a segment connecting a vertex and the midpoint of the opposite side centroid: the point of concurrency for the medians, which is also the point where the triangular shape will balance

Medians and Altitudes of Triangles altitude of a triangle: the perpendicular segment connecting a vertex to the opposite side orthocenter: the point of concurrency for the lines that contain the altitudes of a triangle

Medians and Altitudes of Triangles Core Concepts (page 176 and 177 in Student Journal) Centroid Theorem The medians of a triangle are concurrent at a point (centroid) that is ⅔ the distance from each vertex to the midpoint of the opposite side

Medians and Altitudes of Triangles Orthocenter Theorem The lines that contain the altitudes of a triangle are concurrent at a point (orthocenter).

Medians and Altitudes of Triangles Examples (pages 177 and 178 in Student Journal) P is the centroid of the triangle. Find PN and QP. #1) QN = 33

Medians and Altitudes of Triangles Solution #1) PN = 22, QP = 11

Medians and Altitudes of Triangles Point D is the centroid of the triangle. Find CD and CE. #4) DE = 7

Medians and Altitudes of Triangles Solution #4) CD = 14, CE = 21

Medians and Altitudes of Triangles Find the coordinates of the centroid of the triangle with the given vertices. #6) A(-2, -1), B(1, 8), C(4, -1)

Medians and Altitudes of Triangles Solution #6) (1, 2)

Medians and Altitudes of Triangles Determine the location (inside, on, or outside) of the orthocenter in the triangle. Then find the coordinates of the orthocenter. #9) X(3, 6), Y(3, 0), Z(11, 0)

Medians and Altitudes of Triangles Solution #9) on, (3, 0)

The Triangle Midsegment Theorem I can use the Triangle Midsegment Theorem to find distances.

The Triangle Midsegment Theorem Vocabulary (page 181 in Student Journal) midsegment of a triangle: a segment connecting 2 sides of a triangle at their midpoints

The Triangle Midsegment Theorem Core Concepts (page 181 in Student Journal) Triangle Midsegment Theorem If a segment joins the midpoint of 2 sides of a triangle, then the segment is parallel to the third side and half as long.

The Triangle Midsegment Theorem Examples (pages 182 and 183 in Student Journal) DE is the midsegment of the triangle. Find x. #1)

The Triangle Midsegment Theorem Solution #1) 14

The Triangle Midsegment Theorem Find the perimeter of triangle DEF. #5)

The Triangle Midsegment Theorem Solution #5) 48 units

Indirect Proof and Inequalities in One Triangle I can use the Triangle Inequality Theorem to find possible side lengths of triangles.

Indirect Proof and Inequalities in One Triangle Vocabulary (page 186 in Student Journal) indirect proof: make the assumption that the desired conclusion is false and then show this assumption is logically impossible to prove the original statement true by contradiction

Indirect Proof and Inequalities in One Triangle Core Concepts (pages 186 and 187 in Student Journal) Triangle Longer Side Theorem If 1 side of a triangle is longer than another side, the angle opposite the longer side is larger than the angle opposite the shorter side.

Indirect Proof and Inequalities in One Triangle Larger Angle Theorem If 1 angle of a triangle is larger than another angle, the side opposite the larger angle is longer than the side opposite the smaller angle.

Indirect Proof and Inequalities in One Triangle Inequality Theorem The sum of the lengths of any 2 sides of a triangle is greater than the length of the third side.

Indirect Proof and Inequalities in One Triangle Additional Example (space on pages 186 and 187 in Student Journal) a) Write an indirect proof. Given that line l is not parallel to line k, prove angle 3 and angle 5 are not supplementary.

Indirect Proof and Inequalities in One Triangle Solution a)

Indirect Proof and Inequalities in One Triangle Examples (page 188 in Student Journal) List the angles of triangle in order from least to greatest. #6)

Indirect Proof and Inequalities in One Triangle Solution #6) Angle C, angle A, angle B

Indirect Proof and Inequalities in One Triangle Is it possible to construct a triangle with the given side lengths? Explain. #9) 3, 12, 17

Indirect Proof and Inequalities in One Triangle Solution #9) no, 3 + 12 < 17

Indirect Proof and Inequalities in One Triangle #13) A triangle has 1 side length of 5 units and another side length of 13 units. Describe the possible lengths of the 3 rd side.

Indirect Proof and Inequalities in One Triangle Solution #13) 8 < s < 18

Inequalities in Two Triangles I can solve problems using the Hinge Theorem.

Inequalities in Two Triangles Core Concepts (page 191 in Student Journal) The Hinge Theorem If the 2 sides of one triangle are congruent to the 2 sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger of the included angles.

Inequalities in Two Triangles Converse of the Hinge Theorem If 2 sides of 1 triangle are congruent to 2 sides of another triangle, and the 3 rd side of the 1 st is longer than the 3 rd side of the 2 nd, then the included angle of the 1 st is larger than the included angle of the 2 nd.

Inequalities in Two Triangles Examples (pages 192 and 193 in Student Journal) Complete the statement with <, = or >. #1) BC ___ EF #4) m<A ___ m<D

Inequalities in Two Triangles Solutions #1) < #4) =

Inequalities in Two Triangles Complete the statement with <, = or >. #7) AB ___ AC

Inequalities in Two Triangles Solution #7) >

Inequalities in Two Triangles Write a proof. #10)

Inequalities in Two Triangles Solution #10)

Inequalities in Two Triangles #13) Starting from a point 10 miles north of Crow Valley, a crow flies northeast for 5 miles. Another crow, starting from a point 10 miles south of Crow Valley, flies due west for 5 miles. Which crow is farther from Crow Valley?

Inequalities in Two Triangles Solution #13) the first crow starting north