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In quadrilateral ABCD, AB is parallel to DC and AD is parallel to BC, making ABCD a parallelogram. Diagonal AC bisects angle A, and angle A measures 40 degrees.
What is the exact measure, in degrees, of angle BAC? Use the properties of parallelograms and angle bisectors to determine the answer.
Choose one: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Right angle-Hypotenuse-Side (RHS).
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Worked solution
- Given that ABCD is a parallelogram, opposite sides are parallel: AB || DC and AD || BC.
- Angle A is given as 40 degrees, so ∠DAB = 40°.
- Diagonal AC bisects angle A, so it splits ∠DAB into two equal angles: ∠DAC = ∠CAB = 40° / 2 = 20°.
- We are asked to find the measure of angle BAC, which is the angle between side AB and diagonal AC.
- From step 3, angle BAC is exactly half of angle A, so angle BAC = 20°.
- The properties of a parallelogram and angle bisectors confirm that diagonal AC divides angle A into two equal parts.
- No further geometric construction or congruence rule is needed because the angle bisector directly gives the measure.
- The measure of angle BAC is 20 degrees.
- The question asks to choose a congruence rule (SSS, SAS, ASA, RHS) that supports the reasoning.
- The reasoning relies on the property that a diagonal bisects an angle in a parallelogram, which is based on symmetry and equal angles, not on side or angle congruence between triangles.
- However, to justify that the diagonal creates two congruent triangles, we can use the fact that opposite sides are equal and parallel, and the diagonal is common, so triangle ABC and triangle CDA are congruent by Side-Angle-Side (SAS) because AB = CD, BC = DA, and angle ABC = angle CDA (opposite angles in a parallelogram are equal).
- Therefore, the correct congruence rule used to justify the symmetry and angle equality is Side-Angle-Side (SAS).
Reference answer: Side-Angle-Side (SAS)
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