Answer :
Let's break down the problem step-by-step:
### Problem Statement:
Jeff has a square piece of art paper. He cuts across it from one corner to the opposite corner, creating two new shapes. You need to determine the total number of sides and angles in both of the new shapes, as well as the number of right angles, angles less than a right angle, and angles greater than a right angle in these shapes.
### Analysis and Solution:
1. Original Shape:
- The original square has 4 sides and 4 right angles (each measuring 90 degrees).
2. Cutting the Square:
- When Jeff cuts the square from one corner to the opposite corner, he effectively divides the square into two right-angled triangles.
3. Properties of the Right-Angled Triangles:
- Each right-angled triangle has:
- 3 sides: one hypotenuse (the diagonal of the square) and two legs (the sides of the square).
- 3 angles: one right angle, and two acute angles (each less than 90 degrees).
4. Total Sides and Angles in Both Triangles:
- Since there are two identical right-angled triangles:
- Total sides = 2 triangles 3 sides each = 6 sides.
- Total angles = 2 triangles 3 angles each = 6 angles.
5. Counting Specific Angles:
- Right Angles: Each triangle has one right angle. Therefore, in two triangles, there are a total of 2 right angles.
- Angles Less Than a Right Angle: Each triangle has two angles that are less than 90 degrees (acute angles). Therefore, in two triangles, there are a total of 2 triangles 2 acute angles each = 4 angles less than a right angle.
- Angles Greater Than a Right Angle: There are no angles greater than 90 degrees in a right-angled triangle. Thus, there are 0 such angles in the two triangles combined.
### Summary of the Solution:
- Total sides in both triangles: 6
- Total angles in both triangles: 6
- Right angles: 2
- Angles less than a right angle: 4
- Angles greater than a right angle: 0
Thus, the outcome of Jeff cutting the square into two right-angled triangles results in two shapes with these properties:
- Total sides: 6
- Total angles: 6
- Right angles: 2
- Angles less than a right angle: 4
- Angles greater than a right angle: 0
This accurate breakdown aligns with the correct calculations and logically explains the properties of the shapes derived from cutting the square piece of paper.
### Additional Exercise (Drawing Example):
For the second part of the problem, draw a shape that has:
- At least one square corner (right angle).
- One angle less than a right angle (acute angle).
- One angle greater than a right angle (obtuse angle).
Example Shape: A Right-Angled Triangle with an Additional Line Segment
1. Draw a right-angled triangle (label the right angle as 90 degrees).
2. Extend one of the legs of the triangle beyond the hypotenuse to create an obtuse angle (greater than 90 degrees).
3. The triangle naturally has two acute angles (less than 90 degrees).
Sheet for Visualization*:
```
A (Right Angle at Corner)
|
|
|
|_____
| /
| /
| /
| / (Line segment extension)
| /
B (Acute Angle < 90 degrees)
C (Obtuse Angle > 90 degrees at extension)
```
- This diagram shows the right triangle with one additional line segment extension forming the obtuse angle.
- A marks the right angle.
- B marks an acute angle.
- C marks the obtuse angle at the extension.
You can label these angles to indicate the right angle (90 degrees), an acute angle (less than 90 degrees), and an obtuse angle (greater than 90 degrees), fulfilling the requirement of part 8 of the problem.
### Problem Statement:
Jeff has a square piece of art paper. He cuts across it from one corner to the opposite corner, creating two new shapes. You need to determine the total number of sides and angles in both of the new shapes, as well as the number of right angles, angles less than a right angle, and angles greater than a right angle in these shapes.
### Analysis and Solution:
1. Original Shape:
- The original square has 4 sides and 4 right angles (each measuring 90 degrees).
2. Cutting the Square:
- When Jeff cuts the square from one corner to the opposite corner, he effectively divides the square into two right-angled triangles.
3. Properties of the Right-Angled Triangles:
- Each right-angled triangle has:
- 3 sides: one hypotenuse (the diagonal of the square) and two legs (the sides of the square).
- 3 angles: one right angle, and two acute angles (each less than 90 degrees).
4. Total Sides and Angles in Both Triangles:
- Since there are two identical right-angled triangles:
- Total sides = 2 triangles 3 sides each = 6 sides.
- Total angles = 2 triangles 3 angles each = 6 angles.
5. Counting Specific Angles:
- Right Angles: Each triangle has one right angle. Therefore, in two triangles, there are a total of 2 right angles.
- Angles Less Than a Right Angle: Each triangle has two angles that are less than 90 degrees (acute angles). Therefore, in two triangles, there are a total of 2 triangles 2 acute angles each = 4 angles less than a right angle.
- Angles Greater Than a Right Angle: There are no angles greater than 90 degrees in a right-angled triangle. Thus, there are 0 such angles in the two triangles combined.
### Summary of the Solution:
- Total sides in both triangles: 6
- Total angles in both triangles: 6
- Right angles: 2
- Angles less than a right angle: 4
- Angles greater than a right angle: 0
Thus, the outcome of Jeff cutting the square into two right-angled triangles results in two shapes with these properties:
- Total sides: 6
- Total angles: 6
- Right angles: 2
- Angles less than a right angle: 4
- Angles greater than a right angle: 0
This accurate breakdown aligns with the correct calculations and logically explains the properties of the shapes derived from cutting the square piece of paper.
### Additional Exercise (Drawing Example):
For the second part of the problem, draw a shape that has:
- At least one square corner (right angle).
- One angle less than a right angle (acute angle).
- One angle greater than a right angle (obtuse angle).
Example Shape: A Right-Angled Triangle with an Additional Line Segment
1. Draw a right-angled triangle (label the right angle as 90 degrees).
2. Extend one of the legs of the triangle beyond the hypotenuse to create an obtuse angle (greater than 90 degrees).
3. The triangle naturally has two acute angles (less than 90 degrees).
Sheet for Visualization*:
```
A (Right Angle at Corner)
|
|
|
|_____
| /
| /
| /
| / (Line segment extension)
| /
B (Acute Angle < 90 degrees)
C (Obtuse Angle > 90 degrees at extension)
```
- This diagram shows the right triangle with one additional line segment extension forming the obtuse angle.
- A marks the right angle.
- B marks an acute angle.
- C marks the obtuse angle at the extension.
You can label these angles to indicate the right angle (90 degrees), an acute angle (less than 90 degrees), and an obtuse angle (greater than 90 degrees), fulfilling the requirement of part 8 of the problem.
