Question

Draw two angles that each measure more than \(90^{\circ} .\) Explain how you know they measure more than \(90^{\circ}\).

Short answer

Two obtuse angles can be drawn using a protractor; examples include 120 degrees and 145 degrees.

Step-by-step solution

Understand the properties of angles

To draw two angles that each measure more than 90 degrees, it is essential to know that angles measuring more than 90 degrees are called obtuse angles. An obtuse angle is one that is greater than 90 degrees but less than 180 degrees.

Use a protractor

To accurately draw angles, use a protractor. Place the midpoint of the protractor at the vertex of the angle. Ensure the baseline of the protractor aligns with one of the sides of the angle. This will help in measuring the specific degrees accurately.

Draw the first angle

Using the protractor, measure and draw an angle that is, for example, 120 degrees. Start by drawing a straight line, then mark a point at the 120-degree mark on the protractor from the baseline. Draw a line connecting the vertex to this point.

Draw the second angle

Again, using the protractor, measure and draw another angle, such as 145 degrees. Repeat the same process: draw a baseline, mark the 145-degree point, and draw a line from the vertex to the point.

Verify the angles measure more than 90 degrees

Verify each angle with the protractor, ensuring that they exceed 90 degrees. By confirming that the measured angles are 120 degrees and 145 degrees respectively, they are each confirmed to be obtuse angles.

Key concepts

Angle Measurement

Angle measurement is fundamental in understanding geometry. We measure angles in degrees, which is a way to quantify the rotation needed to bring one line segment into alignment with another. A full circle is divided into 360 degrees. When we measure angles, it generally lies between 0 and 360 degrees.
To determine the size of an angle, you need tools like a protractor. Angles can be categorized based on their measurements:

• Acute angles: less than 90 degrees
• Right angle: exactly 90 degrees
• Obtuse angles: between 90 and 180 degrees
• Straight angle: exactly 180 degrees

For the problem at hand, measuring and understanding angles that are greater than 90 degrees but less than 180 degrees is crucial.

Protractor Use

A protractor is an essential tool for drawing and measuring angles. Here is a straightforward method on how to use one:

• Place the midpoint of the protractor at the vertex of the angle you are measuring or drawing.
• Ensure the baseline of the protractor aligns perfectly with one side of the angle.
• Find your desired angle measurement on the protractor. For obtuse angles, this will be between 90 and 180 degrees.
• Mark a small dot where your chosen degree measurement is located.
• Draw a line connecting the vertex to this dot.

This method ensures accuracy and helps you visualize angles more clearly. By practicing with a protractor, you can improve your angle-measuring skills which are crucial for geometric problems.

Properties of Angles

Understanding the properties of angles helps in identifying and working with them effectively. For obtuse angles, some key properties are:

• Obtuse angles are greater than 90 degrees and less than 180 degrees.
• They appear 'wider' or 'more open' compared to acute and right angles.
• In a triangle, only one angle can be obtuse, as the sum of all angles in any triangle is always 180 degrees.
• Obtuse angles are typically found in various geometric shapes and are essential for certain constructions and proofs.

Recognizing these properties can guide you in drawing, measuring, and verifying angles effectively. For the exercise, knowing these characteristics confirms that the correctly measured angles are indeed obtuse.