Download 5.1 Angles of Triangles

5.1 Angles of Triangles
Classifying Triangles by Sides and by Angles
Classifying Triangles by Sides
Scalene Triangle- Isosceles TriangleNo congruent
At least 2 congruent
sides
sides
Equilateral Triangle3 congruent sides
Classifying Triangles by Angles
Acute Triangle3 acute angles
Right Triangle1 right angle
Obtuse Triangle1 obtuse angle
Equiangular
Triangle3 congruent angles
Example 1: Classifying Triangles by Sides
and Angles
Classify the triangular shape of the support beams in the
diagram by its sides and angles measures.
The triangle has 2 congruent sides, so
it isosceles.
70°
The triangle has 3 acute angles.
55°
It is an acute is an acute isosceles
triangle.
55°
Example 2: Classifying a Triangle in the
Coordinate Plane
Classify ∆𝑂𝑃𝑄 by its sides. Then determine if it is a right
triangle.
Step 1: Use the distance formula to find the
side lengths.
Distance = (𝒙𝟐 − 𝒙𝟏 )𝟐 +(𝒚𝟐 − 𝒚𝟏 )𝟐
𝑶𝑷 =
𝑶𝑸 =
𝑷𝑸 =
(−𝟏 − 𝟎)𝟐 +(𝟐 − 𝟎)𝟐 = 𝟓 ≈ 𝟐. 𝟐
(𝟔 − 𝟎)𝟐 +(𝟑 − 𝟎)𝟐 = 𝟒𝟓 ≈ 𝟔. 𝟕
𝟔 − (−𝟏) 𝟐 + (𝟑 − 𝟐)𝟐 = 𝟓𝟎 ≈ 𝟕. 𝟏
Because no sides are congruent ∆𝑂𝑃𝑄 is scalene.
Example 2: Classifying a Triangle in the
Coordinate Plane
Classify ∆𝑂𝑃𝑄 by its sides. Then determine if it is a right
triangle.
Step 2: Check for right angles by finding the slope
of each line.
𝚫𝒚 𝒚 −𝒚
slope = = 𝟐 𝟏
𝚫𝒙
𝒙𝟐 −𝒙𝟏
𝟐−𝟏
𝑶𝑷 =
= −𝟐
−𝟏 − 𝟎
𝑷𝑸 =
𝟑−𝟐
1
=
𝟔 − (−𝟏)
7
𝟑−𝟎
𝟏
𝑶𝑸 =
=
𝟔−𝟎
𝟐
Since OP and OQ are opposite reciprocals
they are perpendicular. This means that angle
O is 90°.
∆𝑶𝑷𝑸 is a right scalene triangle.
Finding Angle Measure of Triangles
When the sides of a polygon are extended, other angles are formed.
The original angles are the interior angles.
The angles that form linear pairs with the interior angles are the
exterior angles.
Triangle Sum Theorem
The sum of the measures of the interior angles of a
triangle is 180°.
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the
sum of the measures of the two nonadjacent interior angles.
Example 3: Finding an Angle Measure
Find m< JKM.
Using the exterior angle theorem m<J + m<L = m<JKM.
x + 70 = 2x – 5
-x
-x
70 = x – 5
+5
+5
75 = x
m<JKM = 2(75) – 5 = 145°
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary.
Example 4: Modeling with Mathematics
In the painting, the red triangle is a right triangle. The
measure of one acute angle in the triangle is twice the
measure of the other. Find the measure of each acute
angle.
<1 = (2x)° <2 = x°
The corollary to the triangle sum
theorem says that the acute
angles in a right triangle are
complementary.
2x + x = 90
3x = 90
x = 30
1
2
m<1 = 60°
m<2 = 30°
You Try!
1)Find the measure of <1.
5(25) – 10
115°
2) Find the measure of each acute angle.
=64°
=26°
Use Exterior Angle Theorem:
3x + 40 = 5x – 10
50 = 2x
25 = x
<1 + 115 = 180
m<1 = 65°
Use the Corollary to the
Triangle Sum Theorem:
2x + (x – 6) = 90
3x – 6 = 90
3x = 96
x = 32