Angle Relationships - SSAT Middle Level Quantitative

1. What is $x$ if two complementary angles measure $x^\circ$ and $35^\circ$?

   $x=55^\circ$. Subtract the known angle from $90^\circ$ since complementary angles sum to $90^\circ$.

2. What is the sum of the interior angles of a quadrilateral?

   $360^\circ$. A quadrilateral can be divided into two triangles, each with interior angles summing to $180^\circ$.

3. What is the relationship between the base angles of an isosceles triangle?

   The base angles are congruent. In an isosceles triangle, two sides are equal, making the angles opposite them equal.

4. What is $x$ if same-side interior angles measure $x^\circ$ and $64^\circ$ (parallel lines)?

   $x=116^\circ$. Add the known angle to reach $180^\circ$ since same-side interior angles are supplementary with parallel lines.

5. What is $x$ if alternate interior angles measure $x^\circ$ and $101^\circ$ (parallel lines)?

   $x=101^\circ$. Alternate interior angles are congruent with parallel lines, so they have equal measures.

6. What is the measure of a full rotation (one complete turn)?

   $360^\circ$. A full rotation completes one circle around a point.

7. What is the measure of each interior angle of an equilateral triangle?

   $60^\circ$. In an equilateral triangle, all sides are equal, so all angles are equal, dividing the total sum evenly.

8. What is $x$ if a triangle has angles $52^\circ$, $61^\circ$, and $x^\circ$?

   $x=67^\circ$. Subtract the sum of the known angles from $180^\circ$ by the triangle angle sum theorem.

9. What is the relationship between alternate interior angles with parallel lines and a transversal?

   Alternate interior angles are congruent. Alternate interior angles lie on opposite sides of the transversal between parallel lines, making them equal.

10. What is the relationship between supplementary angles?

    They sum to $180^\circ$. Supplementary angles together form a straight angle.

11. What is the exterior angle theorem for a triangle (in degrees)?

    Exterior angle $=$ sum of two remote interior angles. The exterior angle theorem states that an exterior angle equals the sum of the two non-adjacent interior angles.

12. What is $x$ if corresponding angles measure $x^\circ$ and $73^\circ$ (parallel lines)?

    $x=73^\circ$. Corresponding angles are congruent when lines are parallel, so they have equal measures.

13. What is the measure of a straight angle?

    $180^\circ$. A straight angle forms a straight line, equivalent to half of a full rotation.

14. What is the measure of each angle in a square?

    $90^\circ$. A square is a regular quadrilateral, so each interior angle is one-fourth of the total sum.

15. What is the relationship between alternate exterior angles with parallel lines and a transversal?

    Alternate exterior angles are congruent. Alternate exterior angles lie on opposite sides of the transversal outside parallel lines, making them equal.

16. What is the relationship between adjacent angles that form a linear pair?

    They are supplementary (sum to $180^\circ$). Adjacent angles in a linear pair lie on a straight line, so they add up to a straight angle.

17. What is $x$ if a linear pair measures $x^\circ$ and $132^\circ$?

    $x=48^\circ$. Subtract the known angle from $180^\circ$ since a linear pair sums to $180^\circ$.

18. What is the formula for the sum of interior angles of an $n$-gon?

    $(n-2)\cdot 180^\circ$. An $n$-gon can be divided into $n-2$ triangles, each contributing $180^\circ$ to the total sum.

19. What is $x$ if two supplementary angles measure $x^\circ$ and $118^\circ$?

    $x=62^\circ$. Subtract the known angle from $180^\circ$ since supplementary angles sum to $180^\circ$.

20. What is the relationship between vertical angles formed by intersecting lines?

    Vertical angles are congruent (equal). Vertical angles are opposite each other when two lines intersect and share the same vertex, making them equal.

21. What is $x$ if two vertical angles measure $x^\circ$ and $47^\circ$?

    $x=47^\circ$. Vertical angles are congruent, so they have equal measures.

22. What is the relationship between complementary angles?

    They sum to $90^\circ$. Complementary angles together form a right angle.

23. What is the relationship between corresponding angles when two parallel lines are cut by a transversal?

    Corresponding angles are congruent. When parallel lines are cut by a transversal, corresponding angles are in matching positions and thus equal.

24. What is the sum of the interior angles of any triangle?

    $180^\circ$. The triangle angle sum theorem states that the three interior angles of any triangle add up to a straight angle.

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