Geometry #1

A segment, angle, or quantity is compared with itself—for example, AB ≅ AB. Which reason applies?

Reflexive Property — Every figure or quantity is congruent or equal to itself.

If AB ≅ CD and CD ≅ EF, then AB ≅ EF. Which reason applies?

Transitive Property — If one quantity is congruent or equal to a second, and the second is congruent or equal to a third, then the first and third are congruent or equal.

One equal expression is replaced by another equal expression in an equation or proof. Which reason applies?

Substitution — Replace a quantity or expression with one that is equal to it.

Two triangles have already been proven congruent. You now claim that a matching side or angle is congruent. Which reason applies?

Corresponding Parts of Congruent Triangles are Congruent (CPCTC) — Once two triangles are congruent, all corresponding sides and angles are congruent.

Two lines intersect, creating a pair of opposite angles. Why are those angles congruent?

Vertical angles are congruent — Opposite angles formed by intersecting lines have equal measures.

∠1 and ∠2 are both complementary to the same angle, or to congruent angles. Why is ∠1 ≅ ∠2?

If two angles are complements of the same angle (or congruent angles), then they are congruent — Both angles complete the same 90° total.

∠1 and ∠2 are both supplementary to the same angle, or to congruent angles. Why is ∠1 ≅ ∠2?

If two angles are supplements of the same angle (or congruent angles), then they are congruent — Both angles complete the same 180° total.

Two angles are both right angles. Why are they congruent?

All right angles are congruent — Every right angle measures 90°.

In a triangle, two sides are congruent. What proves that the angles opposite those sides are congruent?

In a triangle, angles opposite of congruent sides are congruent — Equal sides in a triangle have equal opposite angles.

A quadrilateral is known to be a parallelogram, rhombus, rectangle, or square. What proves that its opposite angles are congruent?

Opposite angles of a parallelogram / rhombus / rectangle / square are congruent — Each pair of opposite angles has equal measure.

A quadrilateral is known to be a rectangle or square. What proves that all four angles are congruent?

In a rectangle / square, all angles are congruent — All four angles are right angles.

A ray bisects an angle. What proves that the two smaller angles are congruent?

An angle bisector divides an angle into two congruent angles — A bisector makes two equal-angle parts.

Parallel lines are cut by a transversal. Which reason proves that the alternate interior angles are congruent?

Parallel lines cut by a transversal form congruent alternate interior angles — The angles lie between the parallel lines on opposite sides of the transversal.

Parallel lines are cut by a transversal. Which reason proves that the alternate exterior angles are congruent?

Parallel lines cut by a transversal form congruent alternate exterior angles — The angles lie outside the parallel lines on opposite sides of the transversal.

Parallel lines are cut by a transversal. Which reason proves that the corresponding angles are congruent?

Parallel lines cut by a transversal form congruent corresponding angles — The angles occupy matching corners at the two intersections.

Two angles of one triangle are congruent to two angles of another triangle. What proves that the remaining pair of angles is congruent?

If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent — Each triangle's angles total 180°.

Congruent angle measures are added to other congruent angle measures. What can you conclude about the sums?

Congruent angles added to congruent angles form congruent angles — Adding equal angle measures produces equal sums.

Congruent angle measures are subtracted from other congruent angle measures. What can you conclude about the differences?

Congruent angles subtracted from congruent angles form congruent angles — Subtracting equal angle measures from equal angle measures produces equal differences.

Each of two angles is half the measure of a pair of congruent angles. What can you conclude?

Angles that are each half the measure of congruent angles are congruent — Halves of equal angle measures are equal.

Each of two angles is twice the measure of a pair of congruent angles. What can you conclude?

Angles that are each twice the measure of congruent angles are congruent — Doubles of equal angle measures are equal.