To solve this problem, we first need to determine the measure of angle C in triangle ABC using the fact that the sum of the angles in a triangle is always 180 degrees.
Given:
- Measure of angle A = 40 degrees
- Measure of angle B = 70 degrees
We can find angle C as follows:
\[ \text{Measure of angle C} = 180^\circ - 40^\circ - 70^\circ = 70^\circ \]
Therefore:
- Angle A = 40 degrees (smallest angle)
- Angle B = 70 degrees (equal to angle C)
- Angle C = 70 degrees (equal to angle B)
Next, let's consider the implications for the side lengths based on these angles:
1. In any triangle, the side opposite a larger angle is longer than the side opposite a smaller angle.
2. Since angles B and C are equal, sides AB and BC (opposite these angles) are equal.
Let's analyze each statement:
A. **AC = AB**: This is not necessarily true since side AC is opposite 70 degrees (equal to AB), but AC is not necessarily equal to AB without further information implying equal sides.
B. **AB > AC**: Since sides opposite equal angles are equal in length (AB and BC), this statement cannot be true without further information that AB is longer than AC, which the problem doesn't establish.
C. **AC < BC**: Since angle A (40 degrees) is the smallest angle, side BC (opposite angle A) would be the longest if angle C were larger, but as angles B and C are equal, AC cannot be smaller than BC without their direct relationship provided.
D. **AB > BC**: Given that angles B and C are equal, sides AB and BC would also be equal, so this statement is false without an indication that AB is longer.
E. **AB < BC**: Again, since angles B and C are equal, AB = BC.
F. **AC > BC**: Since angles B and C are equal, sides AB and BC are equal. Angle A (40 degrees, smallest angle) indicates AC could be shorter if angle C was longer, without specific indication, general statement AC > BC may not hold while sides are unspecified.
Given that the lengths of sides in a triangle directly correspond to their opposite angles and angle measures rather:
- **AC** and **AB** are opposite 70-degree angles implying equal lengths.
- **BC** and **AB** are on equal angles giving AB and BC are equal lengths, providing more analytical correction based on relational structure **triangle sides AB<>BC.**
By these interpretations:
- Correct answers: among choices B, F can assert AB <> AC, BC aligns A=40 distinguishing whether **AB>** or **BC**.
Select all the correct answers.
In triangle ABC< measure angle A = 40 degrees. and measure B = 70 degrees. Which statements about triangle ABC are true?
A. AC = AB
B. AB > AC
C. AC < BC
D. AB > BC
E. AB < BC
F. AC > BC
1 answer
1 answer