Point C is on the x-axis, so its y-coordinate is 0.
Point C is the projection of A onto the x-axis, so their x-coordinates are the same.
Let's add these coordinates on the diagram.
b
We are given a hint that tells us to find the relationship between angles ∠1, ∠2, and ∠3. Let's start with ∠1 and ∠2.
Angles ∠1 and ∠2, together with the right angle between lines l_1 and l_2, form a straight angle, so their measures add up to 180.
m∠1+90+m∠2=180
Let's now look at triangle â–³ OBD.
Two of the angles of this triangle are ∠2 and ∠3.
According to the Triangle Angle-Sum Theorem, the interior angle measures add up to 180.
m∠3+90+m∠2=180
We now have two angle sums that both give 180.
Let's set them equal to each other and simplify the equation.
This means that ∠1 and ∠3 are congruent.
These are angles of triangles â–³ OBD and â–³ OAC.
Note that these triangles are right-angled, so they have two pairs of congruent angles.
If we choose B so that OA=OB, then triangles â–³ OBD and â–³ OAC also have a congruent side.
According to the Angle-Angle-Side (AAS) Congruence Theorem, the triangles are congruent and hence all corresponding sides are congruent.
Let's see how we can use this to find the coordinates of B and D.
Point D is on the x-axis, so the y-coordinate is 0. Since AC=b and DO is congruent to AC, the x-coordinate of D is - b.
Segment BD is vertical, so the x-coordinate of B is also - b. Since OC=a and DB is congruent to OC, the y-coordinate of B is a.
c
We now have the coordinates of a point on both lines and we know that they intersect at the origin.
Let's use the Slope Formula to find the slope of the lines.
