1. 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

2. 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

3. 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

4. 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

5. 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

6. Medians
CD = 14 D C B A E F

7. End of Section 6.1

8. 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

9. 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.

10. 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

11. 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

12. 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.

13. 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

14. End of Section 6.2

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

16. 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

17. 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.

18. 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

19. End of Section 6.3

20. 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

21. 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

22. 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

23. 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.

24. End of Section 6.4

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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?

33. End of Section 6.5

34. 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

35. 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 =
25 =

36. 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.

37. End of Section 6.6

38. 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

39. 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

40. 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 =

41. 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

42. End of Section 6.7

43. Distance on the Coordinate Plane