geometry

Homework 2 Theorem 36. Congruent angles have congruent supplements.

Proof. Refer to Figure 1. Let ∠BDA and ∠BDC be sup- plementary, ∠EGH and ∠HGF be supplementary, and further suppose we are given ∠BDA ∼= ∠EGH. We will show that ∠BDC ∼= ∠HGF .

Figure 1: Diagram for congruent supplements theorem

Corollary 37. Vertical angles are congruent.

Proof. Refer to Figure 2. Let ←→ AB and

←→ CD be distinct

lines and let X be a point that is on both lines. We wish to show that ∠AXB ∼= ∠CXD.

Figure 2: Vertical angles

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Theorem 38. An angle that is congruent to a right angle is also a right angle.

Proof. Refer to Figure 3. Let ∠BXA be given, with sup- plementary ∠BXC, and let ∠EGH be given, with sup- plementary ∠HGF , and suppose that ∠BXA is a right angle, and ∠BXA ∼= ∠EGH. Then . . .

Figure 3: Weak right angle theo- rem

Theorem 39. If two lines have a transversal which forms alternate interior angles that are congruent, then the two lines are parallel.

Proof. Refer to Figure 4. This will be a proof by contra-

diction. Suppose that lines ←→ AB and

←→ CD are given, with

transversal ←→ EF, with G the point on

←→ AB and

←→ EF, and

H the point on ←→ CD and

←→ EF. Furthermore, suppose that

∠BGH ∼= ∠CHG. We will suppose that lines ←→ AB and

←→ CD are not parallel, and so intersect at point J, which

we may assume is on −−→ AB.

Construct a circle with center H and radius the length of GJ, and let K be the point of intersection of this circle

with −−→ HC. Construct GK.

Figure 4: Alternatie interior an- gles

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Corollary 40. If two lines have a transversal which forms corresponding angles that are congruent, then the two lines are parallel.

Proof.

Corollary 41. Given a line k and a point P not on k, there is a line m such that P is on m and m is parallel to k.

Proof. Refer to Figure 5. Let ←→ AB be given, and point P

be given not on ←→ AB.

Figure 5: Parallel line construc- tion

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