Classifying Triangles
Objectives • Identify the parts of triangles and classify triangles by their parts Notes A triangle is a three-sided polygon.
A C B
Examples Refer to the figure. Triangle ABC is isosceles with AB > AC and AB > BC. Also, XY AB . Name each of the following. 1. sides of the triangle 2. angles of the triangle 3. vertex angle 4. base angles 5. side opposite ∠BCA 6. congruent sides 7. angle opposite AC
A polygon is a closed figure in a plane that is made up of segments, called sides that intersect only at their endpoints, called vertices. ∠BAC or ∠A Sides: AB, AC , BC Angles: ∠ABC or ∠B Vertices: A, B, C ∠ACB or ∠C One way to classify triangles is by their angles. All triangles have at least two acute angles, but the third angle, which is used to classify the triangle, can be acute, right or obtuse.
In an acute triangle, all angles are acute.
72
In an obtuse triangle, one angle is obtuse.
104
30
48 60
46
In a right triangle, one angle is right.
39
An equiangular triangle is an acute triangle in which all angles are congruent.
60
8. Triangle RST is isosceles. Find x,RS, ST, and RT.
x+7
S
x −1
R
90 51
60
60
3x − 5
T
Triangles can also be classified according to the number of congruent sides they have. An equal number of slashes on sides of a triangle indicate that those sides are congruent.
No two sides of a scalene triangle are congruent.
At two sides of an isosceles triangle are congruent.
All sides of an equilateral triangle are congruent.
Like the right triangle, the parts of an isosceles triangle have special names.
vertex angle leg leg
base angle base
base angle
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Examples
9. Triangle PQR is an equilateral triangle. One side measures 2 x + 5 and another side measures x + 35 . Find the length of each side.
10. Given ∆DAR with vertices D ( 2, 6 ) , A ( 4, −5 ) , and A ( −3, 0 ) , use the distance formula to show that ∆DAR is scalene.
Homework Classifying Triangles In the figure, ∆BLM is isosceles with base ML . 1. Identify an acute angle. 2. Name the hypotenuse. 3. Name the vertex angle. 4. Name the side opposite ∠C . 5. Name the angle opposite MB. 6. Name the base angles. 7. Name the vertices of the right triangle. 8. Name the legs of the isosceles triangle. Use the distance formula to classify each triangle by the measures of its sides. 9. ∆PQR with P ( 0, 6 ) , Q ( 3, 6 ) , and R ( 3, 0 ) .
11. Given ∆STU with vertices S ( 2,3) , T ( 4,3) , and U ( 3, −2 ) , use the distance formula to show that ∆STU is isosceles.
10. ∆SUV with S ( −3, −1) , U ( 2,1) , and V ( 2, −3) . 11. ∆KLM with K ( 4, 0 ) , L ( −2, 0 ) , and M (1,5) . 12. Determine the value of r so that a line through the points at (r, 2) and (4, -6) has a slope of 8 − . 3 13. Give a real-world example of two skew lines. 14. The measure of an angle is one-third the measure of its compliment. 15. Find the value of a so that the distance between M(5,6) and N(a, 10) is 5 units. 16. Solve −6 = 5 x + 9 .