a Follow the given steps. Help yourself with a compass.
B
b What do you know from ∠ ABD and ∠ DBC? How does it help you?
C
c Use the Angle Addition Postulate.
A
a See solution.
B
b ∠ JKL and ∠ DBC are complementary angles.
C
c See solution.
Practice makes perfect
a According to the instructions, first we need to draw right angle ABC and point D in the interior. Then we draw line BD.
Next, we need to draw ray KL and the angle JKL in such a way that ∠ JKL ≅ ∠ ABD. To do it, we follow the steps below:
Draw an arbitrary ray KL. Then, with a compass, put the tip on B and draw any arc that intersects both BD and BA.
Without changing the compass setting, put the tip on K and draw an arc that intersects KL.
Then put the compass tip on D, and open it to the distance between A and D. After that, with the same setting, put the compass tip on L and draw an arc that intersects the arc we draw in step 2.
Finally, name the intersection point between the two arcs J. Then draw the ray KJ. We have finished the construction of ∠ JKL which is congruent to ∠ ABD.
c Let's start writing our conjecture in two parts: the given information and what we want to prove.
Given: & ∠ JKL ≅ ∠ ABD
Prove: & ∠ JKL and ∠ DBC are complementary
By the Angle Addition Postulate, we can rewrite m∠ ABC as the sum of the measures of two smaller angles.
m∠ ABC = m∠ ABD + m∠ DBC
Since ∠ ABC is a right angle, m∠ ABC= 90.
90 = m∠ ABD + m∠ DBC
By construction we know that ∠ JKL ≅ ∠ ABD, so m∠ JKL = m∠ ABD. Replacing this into the equation above, we get the following relation.
90 = m∠ JKL + m∠ DBC
This last equation tells us that ∠ JKL and ∠ DBC are complementary angles, and this is what we wanted to prove.
