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Calculate Missing Triangle Angles: Find Angle ACD and Angle BAC
Mathematics
Grade 8 (Junior High School)
Question Content
Solve for each missing value. Find the measure of angle ACD. Find the measure of angle BAC. (Given triangle with angle at A having a 58° angle, angle at B is 60°, point C extended to D forming exterior angle ACD labeled x)
Correct Answer
Angle ACD = 118°, Angle BAC = 62°
Detailed Solution Steps
1
Step 1: Calculate the measure of angle BAC first. Use the triangle angle sum theorem: the sum of interior angles in a triangle is 180°. Let angle BAC = y. So 60° + 58° + y = 180°.
2
Step 2: Solve for y: y = 180° - 60° - 58° = 62°, so angle BAC = 62°.
3
Step 3: Calculate angle ACD using the exterior angle theorem for triangles, which states an exterior angle equals the sum of the two non-adjacent interior angles. So angle ACD = angle ABC + angle BAC = 60° + 58° = 118°. Alternatively, use linear pair: angle ACB + angle ACD = 180°, angle ACB = 62°, so angle ACD = 180° - 62° = 118°.
Knowledge Points Involved
1
Triangle Angle Sum Theorem
This theorem states that the sum of the three interior angles of any triangle is always 180°. It is used to find unknown interior angles when two other angles are known, applicable to all types of triangles (acute, obtuse, right).
2
Exterior Angle Theorem of Triangles
The theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle. It provides a shortcut to calculate exterior angles without first finding the adjacent interior angle.
3
Linear Pair of Angles
A linear pair is a pair of adjacent angles formed when two lines intersect. The sum of the measures of angles in a linear pair is 180°, meaning they are supplementary. This is used to relate an interior angle of a triangle to its adjacent exterior angle formed by extending a side.
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