In the figure shown, l1∥l2. If m∠8=49°, determine the measures of the other seven angles in the figure.
they are all either 49 or 131
review alternate interior/exterior, vertical, consecutive angles
In the figure shown, l_{1}\parallel l_{2}l
1
∥l
2
. If m∠8 = 49°, determine the measures of the other seven angles in the figure.
To determine the measures of the other seven angles in the figure, we will start by applying the properties of parallel lines. Here is the step-by-step solution:
Step 1: Identify corresponding angles:
a. ∠4 and ∠8 are corresponding angles and are congruent. Thus, m∠4 = m∠8 = 49°.
b. ∠1 and ∠5 are also corresponding angles and are congruent. Thus, m∠1 = m∠5.
Step 2: Identify alternate interior angles:
a. ∠4 and ∠5 are alternate interior angles and are congruent. Thus, m∠4 = m∠5.
Step 3: Identify alternate exterior angles:
a. ∠3 and ∠8 are alternate exterior angles and are congruent. Thus, m∠3 = m∠8 = 49°.
b. ∠2 and ∠7 are also alternate exterior angles and are congruent. Thus, m∠2 = m∠7.
Step 4: Identify consecutive interior angles:
a. ∠3 and ∠6 are consecutive interior angles and are supplementary. Thus, m∠3 + m∠6 = 180°.
b. We already know that m∠3 = 49° from step 3a. So, substituting this value, we have 49° + m∠6 = 180°.
c. Solving for m∠6, subtract 49° from both sides: m∠6 = 180° - 49° = 131°.
Step 5: Identify vertical angles:
a. ∠6 and ∠8 are vertical angles and are congruent. Thus, m∠6 = m∠8 = 131°.
Summary:
- ∠1 = ∠5 (corresponding angles)
- ∠2 = ∠7 (alternate exterior angles)
- ∠3 = ∠8 = 49° (alternate exterior angles)
- ∠4 = ∠5 = 49° (corresponding angles)
- ∠6 = ∠8 = 131° (consecutive interior angles)
- ∠7 = ∠2 (alternate exterior angles)
To determine the measures of the other seven angles in the figure, we need to apply some geometric properties and rules.
Let's examine the given information and the figure provided:
- We have two parallel lines, denoted as l1 and l2.
- The angle 8 is given as 49°.
To find the measures of the other angles, we can use the following rules:
1. Corresponding Angles: When a transversal intersects two parallel lines, the corresponding angles are congruent.
2. Alternate Interior Angles: When a transversal intersects two parallel lines, the alternate interior angles are congruent.
3. Co-interior Angles: When a transversal intersects two parallel lines, the co-interior angles are supplementary. That is, they add up to 180°.
Let's label the angles in the figure for future reference:
- Angle 1 (denoted as m∠1)
- Angle 2 (denoted as m∠2)
- Angle 3 (denoted as m∠3)
- Angle 4 (denoted as m∠4)
- Angle 5 (denoted as m∠5)
- Angle 6 (denoted as m∠6)
- Angle 7 (denoted as m∠7)
Now, let's determine the measures of these angles step by step:
1. Angle 1 and Angle 8 are corresponding angles. Therefore, m∠1 = m∠8 = 49°.
2. Angle 1 and Angle 4 are alternate interior angles. Therefore, m∠1 = m∠4.
3. Angle 1 and Angle 4 are co-interior angles. They are supplementary, which means they add up to 180°. Since m∠1 = m∠4, we can write the equation: m∠1 + m∠4 = 180°.
4. Angle 4 and Angle 5 are corresponding angles. Therefore, m∠4 = m∠5.
5. Angle 4 and Angle 5 are alternate interior angles. Therefore, m∠4 = m∠5.
6. Angle 5 and Angle 7 are alternate interior angles. Therefore, m∠5 = m∠7.
7. Angle 5 and Angle 7 are co-interior angles. They are supplementary, which means they add up to 180°. Since m∠5 = m∠7, we can write the equation: m∠5 + m∠7 = 180°.
From these steps, we have determined the relationships and equations for finding the measures of angles in the figure. We will now solve for the unknown angles using the given information and equations.