Lesson 4

1. Lesson 4 Triangle Basics

3. Definition • A triangle is a three-sided figure formed by joining three line segments together at their endpoints. • A triangle has three sides. • A triangle has three vertices (plural of vertex). • A triangle has three angles. 3 2 1

4. Naming a Triangle • Consider the triangle shown whose vertices are the points A, B, and C. • We name this triangle by writing a triangle symbol followed by the names of the three vertices (in any order). C Name A B

5. The Angles of a Triangle • The sum of the measures of the three angles of any triangle is • Let’s see why this is true. • Given a triangle, draw a line through one of its vertices parallel to the opposite side. • Note that because these angles form a straight angle. • Also notice that angles 1 and 4 have the same measure because they are alternate interior angles and the same goes for angles 2 and 5. • So, replacing angle 1 for angle 4 and angle 2 for angle 5 gives 4 5 3 2 1

6. Example • In • What is

7. D C B A Example • In the figure, is a right angle and bisects • If then what is 25 50 90 ? 65 40

8. Angles of a Right Triangle • Suppose is a right triangle with a right angle at C. • Then angles A and B are complementary. • The reason for this is that B A C

9. Exterior Angles • An exterior angle of a triangle is an angle, such as angle 1 in the figure, that is formed by a side of the triangle and an extension of a side. • Note that the measure of the exterior angle 1 is the sum of the measures of the two remote interior angles 3 and 4. To see why this is true, note that 4 2 1 3

10. Classifying Triangles by Angles • An acute triangle is a triangle with three acute angles. • A right triangle is a triangle with one right angle. • An obtuse triangle is a triangle with one obtuse angle. acute triangle right triangle obtuse triangle

11. A C B Right Triangles • In a right triangle, we often mark the right angle as in the figure. • The side opposite the right angle is called the hypotenuse. • The other two sides are called the legs. hypotenuse leg leg

12. Classifying Triangles by Sides • A triangle with three congruent sides is called equilateral. • A triangle with two congruent sides is called isosceles. • A triangle with no congruent sides is called scalene. scalene isosceles equilateral

13. Angles and Sides • If two sides of a triangle are congruent… • then the two angles opposite them are congruent. • If two angles of a triangle are congruent… • then the two sides opposite them are congruent.

14. Equilateral Triangles • Since all three sides of an equilateral triangle are congruent, all three angles must be congruent too. • If we let represent the measure of each angle, then

15. Isosceles Triangles • Suppose is isosceles where • Then, A is called the vertex of the isosceles triangle, and is called the base. • The congruent angles B and C are called the base angles and angle A is called the vertex angle. B A C

16. Example • is isosceles with base • If is twice then what is • Let denote the measure of • Then A x 2x 2x B C

17. Example • In the figure, and • Find • Since is isosceles, the base angles are congruent. So, 25 A 130 D 25 50 110 B 20 C

18. Inequalities in a Triangle • In any triangle, if one angle is smaller than another, then the side opposite the smaller angle is shorter than the side opposite the larger angle. • Also, in any triangle, if one side is shorter than another, then the angle opposite the shorter side is smaller than the angle opposite the longer side.

19. Example • Rank the sides of the triangle below from smallest to largest. • First note that • So, C B A

20. Medians • A median in a triangle is a line segment drawn from a vertex to the midpoint of the opposite side. • An amazing fact about the three medians in a triangle is that they all intersect in a common point. We call this point the centroid of the triangle.

21. Another fact about medians is that the distance along a median from the vertex to the centroid is twice the distance from the centroid to the midpoint. 2x x

22. Example • In the medians are drawn, and the centroid is point G. • Suppose • Find A N 4.5 C G P 4 7 M B

23. Midlines • A midline in a triangle is a line segment connecting the midpoints of two sides. • There are two important facts about a midline to remember: midline x 2x

24. C D E A Example • In D and E are the midpoints of respectively. • If and then find and B

25. A c b a C B The Pythagorean Theorem • Suppose is a right triangle with right angle at C. • The Pythagorean Theorem states that • Here’s another way to state the theorem: label the lengths of the sides as shown. Then

26. leg leg hypotenuse • In words, the Pythagorean Theorem states that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse, or:

27. A C B Example • Suppose is a right triangle with right angle at C.

28. 45 45 45-45-90 Triangles • A 45-45-90 triangle is a triangle whose angles measure • It is a right triangle and it is isosceles. • If the legs measure then the hypotenuse measures • This ratio of the sides is memorized, and if one side of a 45-45-90 triangle is known, then the other two can be obtained from this memorized ratio.

29. Example • In is a right angle and • If then find • First notice that too since the angles must add up to • Then this is a 45-45-90 triangle and so: B 6 ? 45 C A

30. C B A 30-60-90 Triangles • A 30-60-90 triangle is one in which the angles measure • The ratio of the sides is always as given in the figure, which means: • The side opposite the angle is half the length of the hypotenuse. • The side opposite the angle is times the side opposite the angle.

31. B C A Example • In • If find • First note that, since the three angles must add up to • So this is a 30-60-90 triangle.

32. The Converse of the Pythagorean Theorem • Suppose is any triangle where • Then this triangle is a right triangle with a right angle at C. • In other words, if the sides of a triangle measure a, b, and c, and then the triangle is a right triangle where the hypotenuse measures c.

33. 25 24 7 Example • Show that the triangle in the figure with side measures as shown is a right triangle.