Angle Relationships - SSAT Upper Level Quantitative

1. What is $x$ if $x$ and $68^\circ$ form a linear pair?

   $112^\circ$. Subtract $68^\circ$ from $180^\circ$ as angles in a linear pair are supplementary.

2. What is the sum of the measures of the interior angles of a triangle?

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

3. What is the exterior angle if the remote interior angles are $35^\circ$ and $65^\circ$?

   $100^\circ$. Add the remote interior angles according to the exterior angle theorem.

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

   Alternate interior angles are congruent. With parallel lines and a transversal, alternate interior angles are equal by the alternate interior angles theorem.

5. What is $x$ if same-side interior angles are $x$ and $105^\circ$ (parallel lines)?

   $75^\circ$. Subtract $105^\circ$ from $180^\circ$ as same-side interior angles are supplementary with parallel lines.

6. What is $x$ if corresponding angles measure $x$ and $72^\circ$ (parallel lines)?

   $72^\circ$. Corresponding angles are congruent when lines are parallel, so they have equal measures.

7. What is the relationship between same-side (consecutive) interior angles with parallel lines?

   They are supplementary (sum to $180^\circ$). Same-side interior angles add to $180^\circ$ when formed by parallel lines and a transversal, per the consecutive interior angles theorem.

8. What is $x$ if two angles are complementary and their measures are $2x^\circ$ and $5x^\circ$?

   $x=\frac{90}{7}$. Add the expressions and set equal to $90^\circ$, then solve for $x$ since the angles are complementary.

9. What is the measure of each angle in an equilateral triangle?

   $60^\circ$. All angles in an equilateral triangle are equal, and their sum is $180^\circ$, so each is $60^\circ$.

10. What is the relationship between the measures of complementary angles?

    They sum to $90^\circ$. Complementary angles are defined as two angles whose measures add up to $90^\circ$.

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

    Vertical angles are congruent. Vertical angles, formed by intersecting lines, are equal in measure due to their opposite positions.

12. What is $x$ if angles around a point are $110^\circ$, $95^\circ$, $85^\circ$, and $x$?

    $70^\circ$. Subtract the sum of the known angles from $360^\circ$ as angles around a point total $360^\circ$.

13. What is the measure of an angle that is supplementary to a $47^\circ$ angle?

    $133^\circ$. Subtract the given angle from $180^\circ$ since supplementary angles sum to $180^\circ$.

14. What is the value of $x$ if an isosceles triangle has base angles $x$ and vertex angle $40^\circ$?

    $70^\circ$. In an isosceles triangle, base angles are equal, so subtract the vertex angle from $180^\circ$ and divide by 2.

15. 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 equal due to the corresponding angles postulate.

16. What is $x$ if $x$ is vertical to an angle measuring $125^\circ$?

    $125^\circ$. Vertical angles are congruent, so they share the same measure.

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

    Alternate exterior angles are congruent. With parallel lines and a transversal, alternate exterior angles are equal by the alternate exterior angles theorem.

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

    They are supplementary (sum to $180^\circ$). Adjacent angles forming a straight line add up to $180^\circ$ by definition of a linear pair.

19. What is the measure of an angle that is complementary to a $36^\circ$ angle?

    $54^\circ$. Subtract the given angle from $90^\circ$ since complementary angles sum to $90^\circ$.

20. What is the relationship between the measures of supplementary angles?

    They sum to $180^\circ$. Supplementary angles are defined as two angles whose measures add up to $180^\circ$.

21. What is the relationship between an exterior angle of a triangle and the two remote interior angles?

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

22. What is the sum of the measures of angles around a point?

    $360^\circ$. Angles around a point form a full circle, summing to $360^\circ$.

23. What is $x$ if alternate interior angles measure $x$ and $118^\circ$ (parallel lines)?

    $118^\circ$. Alternate interior angles are congruent with parallel lines, so they have equal measures.

24. What is the measure of each interior angle of a square?

    $90^\circ$. A square has four equal right angles, each measuring $90^\circ$, as the sum of interior angles is $360^\circ$.

25. What is $x$ if a triangle has angles $50^\circ$, $60^\circ$, and $x$?

    $70^\circ$. Subtract the sum of the known angles from $180^\circ$ using the triangle angle sum theorem.

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