Congruent triangles | 3rd grade, 4th grade Math Worksheet ... - Free Printable
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Step-by-step solution for: Congruent triangles | 3rd grade, 4th grade Math Worksheet ...
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Show Answer Key & Explanations
Step-by-step solution for: Congruent triangles | 3rd grade, 4th grade Math Worksheet ...
To solve this problem, we need to find which triangles are congruent.
What does congruent mean?
Congruent shapes are exactly the same. They have the same size and the same shape. Even if you turn them or flip them over, they will still fit perfectly on top of each other.
Let's look at each row in the worksheet:
Row 1:
* Look at triangle a. It is a small triangle pointing up.
* Look at triangle b. It looks like the same size and shape as a, just turned slightly. If you imagine picking up triangle a and turning it, it would fit right on top of b.
* Triangle c is much bigger than a and b.
* Triangle d is very flat and wide.
* Result: Triangles a and b are congruent. (The worksheet already has these connected with a line as an example).
Row 2:
* Look at triangle a. It is a right-angled triangle (it has a square corner) standing tall.
* Look at triangle b. It is small and tilted.
* Look at triangle c. It is a large triangle, but it doesn't look like the same shape as a.
* Look at triangle d. This is also a right-angled triangle. It looks like the same size and shape as a, but it is lying on its side. If you rotate triangle a to the right, it would match triangle d perfectly.
* Look at triangle e. It is tiny.
* Look at triangle f. It is medium-sized but shaped differently.
* Look at triangle g. It is long and flat.
* Result: Triangles a and d are congruent.
Row 3:
* Look at triangle a. It is a scalene triangle (all sides different lengths) pointing somewhat to the left.
* Look at triangle b. It looks like the same shape and size as a, but it is flipped upside down. If you flip triangle a over, it matches b.
* Look at triangle c. It is larger than a and b.
* Look at triangle d. It is long and thin.
* Look at triangle e. It is also long and thin, similar to d but maybe not the same. Let's look closer. Actually, looking at d and e, they don't seem to match any others easily. Let's re-examine a and b. Yes, a and b are definitely the same size and shape.
* Let's check the others again. Is there another pair?
* Triangle c is big. No match.
* Triangle d is a long, narrow triangle.
* Triangle e is also a long, narrow triangle. Let's compare d and e. Triangle d has a very sharp angle at the top. Triangle e has a sharp angle at the bottom left. If you rotate triangle d, does it fit on e? They look very similar in shape and size. Let's look really closely at the side lengths. The longest side of d seems to match the longest side of e. The shortest side of d matches the shortest side of e. So d and e are likely congruent too.
* Wait, let me look at the standard answers for this specific worksheet type. Usually, there is one clear pair per section or distinct pairs.
* Let's re-evaluate Row 3 carefully.
* a and b: Same shape, same size. One is rotated/flipped version of the other. Match.
* c: Too big.
* d and e: These two look like they could be congruent. Let's trace them mentally. Triangle d is oriented vertically. Triangle e is oriented horizontally. If I rotate d 90 degrees clockwise, the long side is at the bottom. The short side is on the right. In triangle e, the long side is at the bottom. The short side is on the left. This suggests they might be mirror images. Congruent triangles can be mirror images (reflections). So yes, d and e appear to be congruent as well.
* However, often in these elementary exercises, there is just one primary pair intended per box unless specified otherwise. Let's look at the visual weight. a and b are very clearly the same. d and e are also very clearly the same. Are there any others? No.
* Let's double-check if a matches anything else. No.
* Let's double-check if d matches anything else. No.
* So, in the third box, there are actually two pairs of congruent triangles: a & b AND d & e.
* Let's re-read the first box. It asked "Which triangles are congruent?" and linked a and b.
* Second box: "Which triangles are congruent?" -> a and d.
* Third box: "Which triangles are congruent?" -> a and b are one pair. d and e are another pair.
* Let's look really closely at d and e again.
* Triangle d: Longest side is vertical. Shortest side is top-right. Medium side is bottom-left.
* Triangle e: Longest side is horizontal. Shortest side is bottom-left. Medium side is top-right.
* If I take triangle d and rotate it 90 degrees counter-clockwise: The vertical long side becomes horizontal at the bottom. The top-right short side becomes top-left. The bottom-left medium side becomes bottom-right.
* Now look at e: Long side is bottom. Short side is bottom-left. Medium side is top-right.
* The orientation doesn't match a simple rotation. What about a flip?
* Flip d horizontally: Long side vertical. Short side top-left. Medium side bottom-right. Rotate 90 deg CCW: Long side horizontal bottom. Short side bottom-left. Medium side top-right. This matches e perfectly.
* So yes, d and e are congruent.
* Therefore, the answers are:
* Box 1: a and b (Already done)
* Box 2: a and d
* Box 3: a and b, and d and e
* *Self-Correction*: Sometimes these worksheets only look for one pair. Let's look at the shapes again. Is it possible c matches something? No. Is it possible a matches d? No, a is "fatter". d is "skinnier".
* So, for the final answer, I will list the pairs found.
Summary of Matches:
1. First Group: Triangles a and b are congruent.
2. Second Group: Triangles a and d are congruent.
3. Third Group: Triangles a and b are congruent. Triangles d and e are congruent.
Final Answer:
Box 1: Triangles a and b are congruent.
Box 2: Triangles a and d are congruent.
Box 3: Triangles a and b are congruent, and triangles d and e are congruent.
What does congruent mean?
Congruent shapes are exactly the same. They have the same size and the same shape. Even if you turn them or flip them over, they will still fit perfectly on top of each other.
Let's look at each row in the worksheet:
Row 1:
* Look at triangle a. It is a small triangle pointing up.
* Look at triangle b. It looks like the same size and shape as a, just turned slightly. If you imagine picking up triangle a and turning it, it would fit right on top of b.
* Triangle c is much bigger than a and b.
* Triangle d is very flat and wide.
* Result: Triangles a and b are congruent. (The worksheet already has these connected with a line as an example).
Row 2:
* Look at triangle a. It is a right-angled triangle (it has a square corner) standing tall.
* Look at triangle b. It is small and tilted.
* Look at triangle c. It is a large triangle, but it doesn't look like the same shape as a.
* Look at triangle d. This is also a right-angled triangle. It looks like the same size and shape as a, but it is lying on its side. If you rotate triangle a to the right, it would match triangle d perfectly.
* Look at triangle e. It is tiny.
* Look at triangle f. It is medium-sized but shaped differently.
* Look at triangle g. It is long and flat.
* Result: Triangles a and d are congruent.
Row 3:
* Look at triangle a. It is a scalene triangle (all sides different lengths) pointing somewhat to the left.
* Look at triangle b. It looks like the same shape and size as a, but it is flipped upside down. If you flip triangle a over, it matches b.
* Look at triangle c. It is larger than a and b.
* Look at triangle d. It is long and thin.
* Look at triangle e. It is also long and thin, similar to d but maybe not the same. Let's look closer. Actually, looking at d and e, they don't seem to match any others easily. Let's re-examine a and b. Yes, a and b are definitely the same size and shape.
* Let's check the others again. Is there another pair?
* Triangle c is big. No match.
* Triangle d is a long, narrow triangle.
* Triangle e is also a long, narrow triangle. Let's compare d and e. Triangle d has a very sharp angle at the top. Triangle e has a sharp angle at the bottom left. If you rotate triangle d, does it fit on e? They look very similar in shape and size. Let's look really closely at the side lengths. The longest side of d seems to match the longest side of e. The shortest side of d matches the shortest side of e. So d and e are likely congruent too.
* Wait, let me look at the standard answers for this specific worksheet type. Usually, there is one clear pair per section or distinct pairs.
* Let's re-evaluate Row 3 carefully.
* a and b: Same shape, same size. One is rotated/flipped version of the other. Match.
* c: Too big.
* d and e: These two look like they could be congruent. Let's trace them mentally. Triangle d is oriented vertically. Triangle e is oriented horizontally. If I rotate d 90 degrees clockwise, the long side is at the bottom. The short side is on the right. In triangle e, the long side is at the bottom. The short side is on the left. This suggests they might be mirror images. Congruent triangles can be mirror images (reflections). So yes, d and e appear to be congruent as well.
* However, often in these elementary exercises, there is just one primary pair intended per box unless specified otherwise. Let's look at the visual weight. a and b are very clearly the same. d and e are also very clearly the same. Are there any others? No.
* Let's double-check if a matches anything else. No.
* Let's double-check if d matches anything else. No.
* So, in the third box, there are actually two pairs of congruent triangles: a & b AND d & e.
* Let's re-read the first box. It asked "Which triangles are congruent?" and linked a and b.
* Second box: "Which triangles are congruent?" -> a and d.
* Third box: "Which triangles are congruent?" -> a and b are one pair. d and e are another pair.
* Let's look really closely at d and e again.
* Triangle d: Longest side is vertical. Shortest side is top-right. Medium side is bottom-left.
* Triangle e: Longest side is horizontal. Shortest side is bottom-left. Medium side is top-right.
* If I take triangle d and rotate it 90 degrees counter-clockwise: The vertical long side becomes horizontal at the bottom. The top-right short side becomes top-left. The bottom-left medium side becomes bottom-right.
* Now look at e: Long side is bottom. Short side is bottom-left. Medium side is top-right.
* The orientation doesn't match a simple rotation. What about a flip?
* Flip d horizontally: Long side vertical. Short side top-left. Medium side bottom-right. Rotate 90 deg CCW: Long side horizontal bottom. Short side bottom-left. Medium side top-right. This matches e perfectly.
* So yes, d and e are congruent.
* Therefore, the answers are:
* Box 1: a and b (Already done)
* Box 2: a and d
* Box 3: a and b, and d and e
* *Self-Correction*: Sometimes these worksheets only look for one pair. Let's look at the shapes again. Is it possible c matches something? No. Is it possible a matches d? No, a is "fatter". d is "skinnier".
* So, for the final answer, I will list the pairs found.
Summary of Matches:
1. First Group: Triangles a and b are congruent.
2. Second Group: Triangles a and d are congruent.
3. Third Group: Triangles a and b are congruent. Triangles d and e are congruent.
Final Answer:
Box 1: Triangles a and b are congruent.
Box 2: Triangles a and d are congruent.
Box 3: Triangles a and b are congruent, and triangles d and e are congruent.
Parent Tip: Review the logic above to help your child master the concept of worksheet congruent triangles.
