5.3 use angle bisectors of triangles | PPT - Free Printable

It looks like you're referring to Example 2, which involves a soccer goalie's position relative to the ball and goalposts, forming congruent angles. Although I can't view the image directly, based on your description and typical geometry problems of this nature, I can explain and solve it.

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🔍 Problem Statement (Reconstructed):

A soccer goalie is positioned such that the angles formed between:
- The goalie and the right goalpost (let’s call it point $ R $),
- The goalie and the left goalpost (point $ L $),
- And the ball (point $ B $),

are congruent — meaning ∠$ BGL $ ≅ ∠$ BGR $, where $ G $ is the goalie’s position.

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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🧩 Key Concept: Congruent Angles & Distance

We are told that the angles from the ball to the goalie to each goalpost are congruent:

$$
\angle BGL \cong \angle BGR
$$

This means that the goalie is positioned such that the angles between the ball and each goalpost are equal.

Let’s consider triangle geometry:

- Points: Ball ($ B $), Goalie ($ G $), Left post ($ L $), Right post ($ R $)
- We are given: ∠$ BGL $ = ∠$ BGR $
- The goalie is between the two posts and the ball is in front of the goal.

This situation suggests that the goalie is equidistant in angular measure from both goalposts as seen from the ball.

But the real question is about how far the goalie must move to reach either goalpost.

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✔ Important Insight:

Even if the angles are congruent, the distances from the goalie to each goalpost may differ depending on the position of the goalie.

But here's the key idea:

> If ∠$ BGL $ ≅ ∠$ BGR $, and the goalie is between the ball and the goal line, then the goalie is equidistant from the two goalposts only if they are symmetrically placed.

However, in most cases, if the angles are congruent and the goalie is not centered, then the distance to one post will be greater than the other.

Wait — actually, let's think carefully.

Let’s suppose:

- The goalie is standing somewhere along the goal line.
- The angles from the ball to the goalie to the left and right goalposts are equal: ∠$ BGL $ = ∠$ BGR $

This configuration implies that the goalie is located on the angle bisector of ∠$ LBR $, but more importantly, if the angles at the goalie are equal, then by the Angle-Side-Angle (ASA) or Law of Sines, we can analyze the distances.

But here's a simpler geometric principle:

> In a triangle, if two angles are equal, the opposite sides are equal.

But in this case, we’re dealing with two triangles:
- Triangle $ BGL $
- Triangle $ BGR $

If ∠$ BGL $ = ∠$ BGR $, and the side $ BG $ is common, and assuming the distances $ BL $ and $ BR $ are equal (since the goalposts are symmetric), then the triangles would be congruent — meaning $ GL = GR $.

But wait — unless the goalie is centered, $ GL $ ≠ $ GR $.

So what does "congruent angles" mean?

Let’s interpret it clearly:

> The angle between the ball and the left goalpost as seen from the goalie is equal to the angle between the ball and the right goalpost.

That is:
$$
\angle BGL = \angle BGR
$$

This implies that the goalie sees the ball and the left post at the same angle as the ball and the right post.

Now, using the Law of Sines in triangles $ BGL $ and $ BGR $:

In $ \triangle BGL $:
$$
\frac{BL}{\sin(\angle BGL)} = \frac{BG}{\sin(\angle BLG)}
$$

In $ \triangle BGR $:
$$
\frac{BR}{\sin(\angle BGR)} = \frac{BG}{\sin(\angle BRG)}
$$

But since $ \angle BGL = \angle BGR $, and $ BL = BR $ (assuming symmetric goal), and $ BG $ is shared, then the triangles are congruent, so $ GL = GR $.

Wait — but this would only be true if the goalie is centered.

But if the goalie is not centered, and still forms equal angles, then something else is going on.

Actually, here's a better way:

> If the angles from the goalie to the ball and each goalpost are equal, then the goalie is equidistant from the two goalposts only if the setup is symmetric.

But in reality, equal angles do not necessarily imply equal distances.

Let’s use a real-world analogy:

Imagine the goalie is closer to the left goalpost. Then, the angle between the ball and the right goalpost would be larger because the right post is farther away. But the problem says the angles are congruent.

So for the angles to be equal, the goalie must be equidistant from both goalposts in terms of angular spread.

But here's the key insight:

> If the angles from the goalie to the ball and each goalpost are equal, then the goalie is on the perpendicular bisector of the segment joining the two goalposts — meaning the goalie is centered.

Therefore, the goalie is equidistant from both goalposts.

Thus, the distance to the left and right goalposts is the same.

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

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❓ Wait — But the Question Asks: Which Way Does the Goalie Have to Move Farther?

If the angles are congruent, and the goalie is centered, then the distances are equal → no difference.

But maybe the image shows the goalie off-center, yet forming congruent angles? That seems impossible unless the ball is also off-center.

Ah! Here’s a critical realization:

> Congruent angles do not mean the goalie is equidistant unless the setup is symmetric.

But let’s suppose the goalie is not centered — say, closer to the left post.

Then, the angle from the goalie to the right post would be larger than to the left post — unless the ball is also positioned asymmetrically.

So if the angles are equal, and the goalposts are symmetric, then the goalie must be centered.

Hence, the distances to both posts are equal.

But the problem might be testing which direction requires more movement based on angular congruence.

Wait — perhaps the key is this:

> The goalie’s position forms congruent angles with the ball and each goalpost.

That is, ∠$ BGL $ = ∠$ BGR $

Now, consider the path the goalie must take to block a shot.

The distance the goalie must run is the straight-line distance from their current position to the goalpost.

So, if the goalie is closer to one post, they have to move less to reach it.

But if the angles are congruent, and the ball is fixed, then the goalie is likely centered.

But let’s suppose the goalie is not centered, but still forms equal angles.

Is that possible?

Yes — only if the ball is not on the centerline.

For example, if the ball is to the right of the center, and the goalie is closer to the left post, then the angle to the right post might be smaller.

But to make the angles equal, the goalie must adjust.

But in fact, if ∠BGL = ∠BGR, then by symmetry, the goalie must be on the angle bisector of ∠LBR.

And if the goal is symmetric, then the angle bisector passes through the center of the goal.

So the goalie is centered.

Therefore, the distances to both posts are equal.

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

But the question asks: "Will the goalie have to move farther to block a shot toward the right goalpost $ R $ or the left goalpost $ L $?"

Answer: Neither — they have to move the same distance.

But maybe the image shows the goalie off-center, yet angles are congruent?

That would contradict standard geometry unless the ball is not symmetric.

Wait — perhaps the angles are not at the goalie, but between the ball and the goalposts?

Let me re-read:

> “A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown.”

Possibly, it means:
- The angle between the ball and the left goalpost (from the goalie) equals the angle between the ball and the right goalpost.

That is: ∠$ BGL $ = ∠$ BGR $

Then, again, this implies symmetry.

So unless the goalposts are not symmetric, the goalie must be centered.

Therefore, the distances $ GL $ and $ GR $ are equal.

So the goalie moves the same distance to either post.

But maybe the problem is asking which direction the goalie must move — left or right — and which is farther?

But if angles are congruent, and the goalie is centered, then both directions are equal.

So answer: The goalie has to move the same distance to block a shot toward either goalpost.

But perhaps the image shows the goalie closer to one side, yet angles are congruent — which is only possible if the ball is not in the center.

But in that case, the angles being equal would imply the goalie is not closer to one post.

Wait — let’s suppose the goalie is closer to the left post.

Then:
- Distance $ GL < GR $
- So the angle $ BGL $ (from goalie to ball to left post) would be smaller than $ BGR $, because the left post is closer.

But the problem says the angles are congruent.

So for the angles to be equal, the goalie must be equidistant from both posts.

Therefore, GL = GR

So the goalie must move the same distance to block either shot.

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✔ Final Answer:

> The goalie has to move the same distance to block a shot toward the right goalpost $ R $ or the left goalpost $ L $, because the congruent angles indicate the goalie is equidistant from both posts.

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📌 Summary:

- Congruent angles from the goalie to the ball and each goalpost imply symmetric positioning.
- This means the goalie is centered between the goalposts.
- Therefore, the distances to both goalposts are equal.
- Hence, the goalie must move the same distance to block a shot toward either post.

✔ Answer: The goalie has to move the same distance to block a shot toward either goalpost.