Geometry: Angles and Triangles

Grade 7 · Math · Study Guide

Explore angle relationships, triangle properties, and how to find missing angle measures using geometric rules.

Angles are measured in degrees. A right angle is exactly 90°, an acute angle is less than 90°, and an obtuse angle is between 90° and 180°. Complementary angles sum to 90°, and supplementary angles sum to 180°.

Vertical angles are formed when two lines intersect. They are always equal. Adjacent angles along a straight line are supplementary. Angles in a triangle always sum to 180° — this is the Triangle Angle Sum Theorem.

Triangles can be classified by their angles: an acute triangle has all angles less than 90°, a right triangle has one 90° angle, and an obtuse triangle has one angle greater than 90°. Triangles can also be classified by side lengths: equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal).

In an isosceles triangle, the base angles (opposite the equal sides) are always equal. In an equilateral triangle, every angle is 60°.

The Triangle Inequality Theorem states that the sum of any two side lengths must be greater than the third side. For example, sides of 3, 4, and 8 cannot form a triangle because 3 + 4 = 7, which is not greater than 8. Use these theorems to find missing angles and determine if triangles can exist.

Key terms

Complementary angles
Two angles whose measures add up to exactly 90°
Supplementary angles
Two angles whose measures add up to exactly 180°
Vertical angles
Opposite angles formed by two intersecting lines; they are always equal
Triangle Angle Sum Theorem
The sum of the three interior angles of any triangle is always 180°
Triangle Inequality Theorem
The sum of any two side lengths of a triangle must be greater than the third side
Isosceles triangle
A triangle with exactly two equal sides; the base angles (opposite the equal sides) are also equal

Flashcards

Acute angle
An angle measuring less than 90°
Right angle
An angle measuring exactly 90°; marked with a small square in diagrams
Obtuse angle
An angle measuring between 90° and 180°
Straight angle
An angle measuring exactly 180°; forms a straight line
Finding a missing triangle angle
Subtract the sum of the two known angles from 180°
Isosceles triangle base angles
If two sides are equal, the angles opposite those sides are also equal
Scalene triangle
A triangle where all three sides and all three angles have different measures
Triangle Inequality check
Add the two shorter sides. If their sum > the longest side, a triangle can be formed.

Questions

  1. A triangle has angles of 45° and 75°. What is the measure of the third angle?

    1. 50°
    2. 60°
    3. 70°
    4. 80°
    Show answer

    Answer: 60°

    180° - 45° - 75° = 60°. The three angles of any triangle must sum to 180°.

  2. Vertical angles formed by two intersecting lines are always equal.

    Show answer

    Answer: True

    Vertical angles are the non-adjacent angles formed when two lines cross. They are always congruent (equal).

  3. Two complementary angles have measures of 35° and ___°.

    Show answer

    Answer: 55

    Complementary angles sum to 90°. So 90° - 35° = 55°.

  4. Which set of side lengths could form a triangle?

    1. 2, 3, 6
    2. 5, 5, 10
    3. 4, 7, 9
    4. 1, 2, 4
    Show answer

    Answer: 4, 7, 9

    Check 4 + 7 = 11 > 9 ✓, 4 + 9 = 13 > 7 ✓, 7 + 9 = 16 > 4 ✓. All three pairs satisfy the Triangle Inequality Theorem.

  5. An equilateral triangle has three angles that each measure 60°.

    Show answer

    Answer: True

    An equilateral triangle has all sides equal, so all angles are equal too. 180° ÷ 3 = 60° each.

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