More About Triangles § 6.1 Medians § 6.2 Altitudes and Perpendicular Bisectors § 6.3 Angle Bisectors of Triangles § 6.4 Isosceles Triangles § 6.5 Right Triangles § 6.6 The Pythagorean Theorem § 6.7 Distance on the Coordinate Plane

Vocabulary What You'll Learn Medians What You'll Learn You will learn to identify and construct medians in triangles 1) ______ 2) _______ 3) _________ Vocabulary median centroid concurrent

the ________ of the side __________________. vertex midpoint Medians In a triangle, a median is a segment that joins a ______ of the triangle and the ________ of the side __________________. vertex midpoint opposite that vertex C B A D F E The medians of ΔABC, AD, BE, and CF, intersect at a common point called the ________. centroid When three or more lines or segments meet at the same point, the lines are __________. concurrent

and the length of the segment from the centroid to the midpoint. Medians There is a special relationship between the length of the segment from the vertex to the centroid and the length of the segment from the centroid to the midpoint. C B A E F D

The length of the segment from the vertex to the centroid is Medians Theorem 6 - 1 The length of the segment from the vertex to the centroid is _____ the length of the segment from the centroid to the midpoint. twice 2x x When three or more lines or segments meet at the same point, the lines are __________. concurrent

Medians CD = 14 D C B A E F

End of Section 6.1

Altitudes and Perpendicular Bisectors What You'll Learn You will learn to identify and construct _______ and __________________ in triangles. altitudes perpendicular bisectors 1) ______ 2) __________________ Vocabulary altitude perpendicular bisector

Altitudes and Perpendicular Bisectors In geometry, an altitude of a triangle is a ____________ segment with one endpoint at a ______ and the other endpoint on the side _______ that vertex. perpendicular vertex opposite C A B D The altitude AD is perpendicular to side BC.

Altitudes and Perpendicular Bisectors Constructing an altitude of a triangle 2) Place the compass point on B and draw an arc that intersects side AC in two points. Label the points of intersection D and E. 3) Place the compass point at D and draw an arc below AC. Using the same compass setting, place the compass point on E and draw an arc to intersect the one drawn. 4) Use a straightedge to align the vertex B and the point where the two arcs intersect. Draw a segment from the vertex B to side AC. Label the point of intersection F. 1) Draw a triangle like ΔABC B A C D E

Altitudes and Perpendicular Bisectors An altitude of a triangle may not always lie inside the triangle. Altitudes of Triangles acute triangle right triangle obtuse triangle The altitude is _____ the triangle The altitude is _____ of the triangle The altitude is _______ the triangle inside a side out side

Altitudes and Perpendicular Bisectors Another special line in a triangle is a perpendicular bisector. A perpendicular line or segment that bisects a ____ of a triangle is called the perpendicular bisector of that side. side Line m is the perpendicular bisector of side BC. m A altitude B D C D is the midpoint of BC.

Altitudes and Perpendicular Bisectors In some triangles, the perpendicular bisector and the altitude are the same. The line containing YE is the perpendicular bisector of XZ. Y E YE is an altitude. E is the midpoint of XZ. X Z

End of Section 6.2

Angle Bisectors of Triangles What You'll Learn You will learn to identify and use ____________ in triangles. angle bisectors 1) ___________ Vocabulary angle bisector

Angle Bisectors of Triangles Recall that the bisector of an angle is a ray that separates the angle into two congruent angles. Q R P S

Angle Bisectors of Triangles An angle bisector of a triangle is a segment that separates an angle of the triangle into two congruent angles. One of the endpoints of an angle bisector is a ______ of the triangle, vertex and the other endpoint is on the side ________ that vertex. opposite A B D C Just as every triangle has three medians, three altitudes, and three perpendicular bisectors, every triangle has three angle bisectors.

Angle Bisectors of Triangles Special Segments in Triangles Segment Type Property altitude perpendicular bisector angle bisector line segment line ray line segment line segment from the vertex, a line perpendicular to the opposite side bisects the side of the triangle bisects the angle of the triangle

End of Section 6.3

Vocabulary What You'll Learn Isosceles Triangles What You'll Learn You will learn to identify and use properties of _______ triangles. isosceles 1) _____________ 2) ____ 3) ____ Vocabulary isosceles triangle base legs

The congruent sides are called ____. legs Isosceles Triangles Recall from §5-1 that an isosceles triangle has at least two congruent sides. The congruent sides are called ____. legs The side opposite the vertex angle is called the ____. base In an isosceles triangle, there are two base angles, the vertices where the base intersects the congruent sides. vertex angle leg base angle base

triangle are congruent, then the angles opposite those sides Isosceles Triangles Theorem 6-2 Isosceles Triangle 6-3 A B C If two sides of a triangle are congruent, then the angles opposite those sides are congruent. The median from the vertex angle of an isosceles triangle lies on the perpendicular bisector of the base and the angle bisector of the vertex angle. A B C D

triangle are congruent, then the sides opposite those angles Isosceles Triangles Theorem 6-4 Converse of Isosceles Triangle A B C If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Theorem 6-5 A triangle is equilateral if and only if it is equiangular.

End of Section 6.4

Vocabulary What You'll Learn Right Triangles What You'll Learn You will learn to use tests for _________ of ____ triangles. congruence right 1) _________ 2) ____ Vocabulary hypotenuse legs

In a right triangle, the side opposite the right angle is called the Right Triangles In a right triangle, the side opposite the right angle is called the _________. hypotenuse The two sides that form the right angle are called the ____. legs hypotenuse leg leg

Right Triangles Recall from Chapter 5, we studied various ways to prove triangles to be congruent: In §5-5, we studied two theorems A B C R S T SSS and A B C R S T SAS

Right Triangles Recall from Chapter 5, we studied various ways to prove triangles to be congruent: In §5-6, we studied two theorems R S T A B C ASA and R S T A B C AAS

SAS The theorems mentioned in Chapter 5, were for ALL triangles. Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-6 LL Theorem If two legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. A C B D F E SAS same as

AAS The theorems mentioned in Chapter 5, were for ALL triangles. Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-7 HA Theorem If ______________ and an (either) __________ of one right triangle are congruent to the __________ and _________________ of another right angle, then the triangles are congruent. the hypotenuse acute angle hypotenuse corresponding angle A C B D F E AAS same as

ASA The theorems mentioned in Chapter 5, were for ALL triangles. Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Theorem 6-6 LA Theorem If one (either) ___ and an __________ of a right triangle are congruent to the ________________________ of another right triangle, then the triangles are congruent. leg acute angle corresponding leg and angle A C B D F E ASA same as

ASS The theorems mentioned in Chapter 5, were for ALL triangles. Right Triangles The theorems mentioned in Chapter 5, were for ALL triangles. So, it should make perfect sense that they would apply to right triangles as well. Postulate 6-1 HL Postulate If the hypotenuse and a leg on one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. A C B D F E ASS Theorem?

End of Section 6.5

Vocabulary What You'll Learn Pythagorean Theorem What You'll Learn You will learn to use the __________ Theorem and its converse. Pythagorean 1) _________________ 2) _______________ * 3) _______ Vocabulary Pythagorean Theorem Pythagorean triple converse

If ___ measures of the sides of a _____ triangle are known, the Pythagorean Theorem If ___ measures of the sides of a _____ triangle are known, the ___________________ can be used to find the measure of the third ____. two right Pythagorean Theorem side c a b A _________________ is a group of three whole numbers that satisfies the equation c2 = a2 + b2, where c is the measure of the hypotenuse. Pythagorean triple 3 5 4 52 = 32 + 42 25 = 9 + 16

lengths of the legs __ and __. c a b Pythagorean Theorem Theorem 6-9 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse __, is equal to the sum of the squares of the lengths of the legs __ and __. c a b c a b Theorem 6-10 Converse of the Pythagorean Theorem If c is the measure of the longest side of a triangle, a and b are the lengths of the other two sides, and c2 = a2 + b2, then the triangle is a right angle.

End of Section 6.6

Distance on the Coordinate Plane What You'll Learn You will learn to find the ______________________on the coordinate plane. distance between two points Nothing new! You learned this in Algebra I. Vocabulary

Distance on the Coordinate Plane Hands-On! y x C(2, 3) 1) On grid paper, graph A(-3, 1) and C(2, 3). A(-3, 1) 2) Draw a horizontal segment from A and a vertical segment from C. B(2, 1) 3) Label the intersection B and find the coordinates of B. QUESTIONS: What is the measure of the distance between A and B? (x2 – x1) = 5 What is the measure of the distance between B and C? (y2 – y1) = 2 What kind of triangle is ΔABC? right triangle If AB and BC are known, what theorem can be used to find AC? Pythagorean Theorem What is the measure of AC? 29 ≈ 5.4

Distance on the Coordinate Plane Theorem 6-11 Distance Formula If d is the measure of the distance between two points with coordinates (x1, y1) and (x2, y2), y x A(x1, y1) B(x2, y2) d then d =

Find the distance between each pair of points Find the distance between each pair of points. Round to the nearest tenth, if necessary. 5 4.5

End of Section 6.7

Distance on the Coordinate Plane