EXAMPLE 1 Use the Angle Bisector Theorems SOLUTION Because JG FG ... - Free Printable

Let's solve this real-world problem step by step using geometry.

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Problem Statement:

A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. The question is:

> Will the goalie have to move farther to block a shot toward the right goalpost R or the left goalpost L?

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Given:

- The goalie is at point G (not labeled, but implied to be near the center of the goal).
- The ball is at point B.
- The left goalpost is L, and the right goalpost is R.
- The angles formed from the ball (B) to the goalie (G) to each goalpost are congruent:
- ∠BGL ≅ ∠BGR

This means:
- The angle between the ball, the goalie, and the left post is equal to the angle between the ball, the goalie, and the right post.

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Goal:

Determine whether the goalie must move farther to block a shot toward R or L.

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Step-by-Step Analysis:

1. Understand the Geometry:
- We're dealing with two triangles:
- Triangle BGL (Ball → Goalie → Left post)
- Triangle BGR (Ball → Goalie → Right post)
- It's given that ∠BGL ≅ ∠BGR — these are the angles at the goalie.
- Also, the segment BG is common to both triangles.
- The sides GL and GR are distances from the goalie to the goalposts.

2. Key Insight:
- Since the angles at the goalie (∠BGL and ∠BGR) are congruent, and BG is shared, we can use triangle congruence or isosceles triangle properties.

3. Use the Angle-Side-Angle (ASA) or Isosceles Rule?
- Wait — we don't know if the sides are equal yet.
- But here's the crucial idea: If the angles from the ball to the goalie to each post are equal, and the goalie is positioned such that those angles are symmetric, then the goalie is equidistant from both posts only if the setup is symmetric.

However, in this case, the key is not symmetry, but rather the distance the goalie must travel to reach either post.

4. What does "move farther" mean?
- The goalie must move from their current position to either L or R to block the shot.
- So, we compare the distances GL and GR.

5. Apply the Law of Sines or Geometric Reasoning:
Let’s consider triangle BGL and BGR.

Given:
- ∠BGL = ∠BGR (given: congruent angles)
- Side BG is common
- But we don’t know anything about the other angles or sides yet.

However, there's a better way: if the angles at G are equal, and BG is common, but the positions of L and R are fixed, then the distances GL and GR depend on where the goalie is.

6. But here’s the critical observation:
- The diagram shows yellow lines from the ball (B) to the goalie (G), and then to the posts L and R.
- The angles between BG and GL, and BG and GR are marked as equal.
- This implies that the goalie is located such that the angular separation from the ball to each post is the same.

That is, the goalie sees the ball and each goalpost at the same angle.

7. Now apply the concept of angle bisector or symmetry**:
- If the angles ∠BGL and ∠BGR are equal, and the goalie is inside the goal, then the goalie lies along the angle bisector of ∠LBR (the angle from left post to ball to right post).
- But more importantly, if the angles at G are equal, and the ball is fixed, then the distances from the goalie to the posts may not be equal unless the setup is symmetric.

8. Wait — but here's a key geometric fact:
In triangle BGL and BGR:
- ∠BGL = ∠BGR (given)
- BG is common
- But we don’t know if the angles at L and R are equal.

So we cannot conclude that GL = GR just from this.

However, look at the diagram: the yellow lines form two triangles with equal angles at G, and the goalie appears to be centered between the posts.

But let's think differently.

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The Real Key: Use of the Converse of the Isosceles Triangle Theorem

If two angles in a triangle are equal, then the opposite sides are equal.

But here, we have two separate triangles.

Instead, consider this:

- From the ball B, the goalie G sees the left post L and the right post R at equal angles.
- That is, ∠BGL = ∠BGR.

Now, suppose we draw lines from G to L and G to R.

Then, in triangles BGL and BGR:
- ∠BGL = ∠BGR (given)
- BG is common
- But unless we know more, we can't say the triangles are congruent.

However, if the goalie is equidistant from both posts, then GL = GR, and the angles would be equal only if the setup is symmetric.

But the problem states that the angles are congruent — so perhaps the goalie is positioned such that the angular view to both posts is the same.

But does that imply equal distance?

No — not necessarily.

But here's the real insight:

> If the goalie sees the ball and the two goalposts forming equal angles, and the goalie is on the perpendicular bisector of LR, then GL = GR.

But the problem doesn't say that.

Wait — look at the diagram again.

In the image:
- The yellow lines go from B to G, then to L and R.
- The angles at G between BG and GL, and BG and GR are marked as equal.
- The goalie is closer to the center of the goal.

So, since the angles are equal, and BG is common, and the goalposts are symmetrically placed, then the goalie is equidistant from both posts.

But wait — is that true?

Actually, no. The equality of angles at G does not imply equal distances unless additional conditions hold.

But here’s a better approach:

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Use the Law of Sines:

In triangle BGL:
\[
\frac{GL}{\sin(\angle GB L)} = \frac{BG}{\sin(\angle GLB)}
\]

In triangle BGR:
\[
\frac{GR}{\sin(\angle GBR)} = \frac{BG}{\sin(\angle GRB)}
\]

But we don’t know the other angles.

Alternatively, consider this:

> If the angles at G are equal, and the goalie is positioned such that the path from B to G to L and B to G to R make equal angles, then the goalie is on the angle bisector of ∠LBR.

And in that case, if the goalposts are symmetric, then the goalie is equidistant from both posts.

But the problem says “forms congruent angles,” which likely means:

- ∠BGL = ∠BGR

That is, the angle between the ball and the left post from the goalie's viewpoint equals the angle between the ball and the right post.

But that doesn’t directly give us distance.

Wait — think about it from the goalie's perspective.

Suppose the goalie is at G, and the ball is at B.

The angle between the direction to the ball and the direction to the left post is equal to the angle between the direction to the ball and the direction to the right post.

That means the goalie is equally angled to both posts relative to the ball.

This suggests that the goalie is on the perpendicular bisector of the line segment joining the two posts, if the ball is also symmetric.

But unless the ball is on the axis of symmetry, the goalie might not be equidistant.

But in this diagram, the ball appears to be centered relative to the goal.

Moreover, the goalposts are symmetric, and the goalie is positioned centrally.

Therefore, the congruent angles suggest symmetry.

Hence, the distances GL and GR are equal.

So, the goalie has to move the same distance to block a shot to either post.

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✔ Final Answer:

No, the goalie does not have to move farther to block a shot toward either goalpost.

Because the angles formed are congruent and the setup is symmetric, the goalie is equidistant from both goalposts.

> Therefore, the goalie has to move the same distance to block a shot toward R or L.

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🔍 Explanation Summary:

- Congruent angles at the goalie (∠BGL ≅ ∠BGR) indicate symmetry.
- Combined with the symmetric placement of the goalposts and the ball’s position, this implies the goalie is equidistant from both posts.
- Thus, the distance to L and R is the same.
- So, the goalie must move the same distance to block either shot.

✔ Answer: The goalie has to move the same distance to block a shot toward R or L.