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Warm UpClassify each angle as acute, obtuse, or right.
1. 2.
3.
4. If the perimeter is 47, find x and the lengths of the three sides.
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Triangle Angles
Section 3.4
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Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths.
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B
AC
AB, BC, and AC are the sides of ABC.
A, B, C are the triangle's vertices.
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Acute Triangle
Three acute angles
Triangle Classification By Angle Measures
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Equiangular Triangle
Three congruent acute angles
Triangle Classification By Angle Measures
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Angle Relationships in TrianglesTriangles
Right Triangle
One right angle
Triangle Classification By Angle Measures
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Obtuse Triangle
One obtuse angle
Triangle Classification By Angle Measures
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Classify BDC by its angle measures.
Example 1A: Classifying Triangles by Angle Measures
Classify ABD by its angle measures.
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Classify FHG by its angle measures.
Example 1
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Equilateral Triangle
Three congruent sides
Triangle Classification By Side Lengths
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Isosceles Triangle
At least two congruent sides
Triangle Classification By Side Lengths
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Scalene Triangle
No congruent sides
Triangle Classification By Side Lengths
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Angle Relationships in TrianglesTriangles
Remember!When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.
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Angle Relationships in TrianglesTriangles
Classify EHF by its side lengths.
Example 2A: Classifying Triangles by Side Lengths
Classify EHG by its side lengths.
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Classify ACD by its side lengths.
Example 2
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Find the side lengths of JKL.
Example 3: Using Triangle Classification
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Find the side lengths of equilateral FGH.
Example 3
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Angle Relationships in TrianglesTriangles
Lesson Review
Classify each triangle by its angles and sides.
1. MNQ
2. NQP
3. MNP
4. Find the side lengths of the triangle.
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Angle Relationships in TrianglesTriangles
Warm Up
1. Find the measure of exterior DBA of BCD, if mDBC = 30°, mC= 70°, and mD = 80°.
2. What is the complement of an angle with measure 17°?
3. How many lines can be drawn through N parallel to MP? Why?
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Angle Relationships in TrianglesTriangles
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Angle Relationships in TrianglesTriangles
An auxiliary line is a line that is added to a figure to aid in a proof.
An auxiliary line used in the Triangle Sum
Theorem
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Angle Relationships in TrianglesTriangles
Example 1A: Application
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A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.
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The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?
Example 2a
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One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?
Example 2: Finding Angle Measures in Right Triangles
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An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side.
Interior
Exterior
4 is an exterior angle.
3 is an interior angle.
The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure.
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Angle Relationships in TrianglesTriangles
Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.
Interior
Exterior
3 is an interior angle.
4 is an exterior angle.
The remote interior angles of 4 are 1 and 2.
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Find mB.
Example 3: Applying the Exterior Angle Theorem
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Find mACD.
Example 3
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Example 4
Find mP and mT.
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Find mK and mJ.
Example 4: Applying the Third Angles Theorem
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Lesson Review: Part I
1. The measure of one of the acute angles in a right triangle is 56 °. What is the measure of the other acute angle?
2. Find mABD. 3. Find mN and mP.
2 3
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Lesson Review: Part II
4. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store?