Question

In the given figure, \(\overline{\mathrm{DE}} \mid 1 \overline{\mathrm{AC}}\). Find the value of \(\mathrm{x}\). (1) 1 (2) 2 (3) 3 (4) 4

Short answer

Question: In the given figure, \(\overline{\mathrm{DE}} \mid \overline{\mathrm{AC}}\) and \(\overline{\mathrm{AB}}\) is a transversal. If angle 1 is \(1^{\circ}\), find the value of x. Answer: The value of x is \(179^{\circ}\).

Step-by-step solution

Identify the parallel lines and transversals

In the given figure, the two parallel lines are \(\overline{\mathrm{DE}}\) and \(\overline{\mathrm{AC}}\). The transversal line is \(\overline{\mathrm{AB}}\) which cuts both parallel lines.

Identify the angle relationships between the parallel lines and the transversal

Since the lines are parallel and cut by a transversal, we can use the alternate interior angle property. This states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

Use the alternate interior angle property to set up an equation

Let the angle formed between \(\overline{\mathrm{AB}}\) and \(\overline{\mathrm{AC}}\) be \(\alpha\) (denoted by the arc symbol on the diagram, and angle formed between \(\overline{\mathrm{AB}}\) and \(\overline{\mathrm{DE}}\) be x. Since DE is parallel to AC and AB is a transversal, we can say that angle \(\alpha\) and angle x are alternate interior angles, which means they are congruent. Using this property, we can set up the equation: $$ \alpha = x $$

Evaluate the angle \(\alpha\)

Angle \(\alpha\) can be evaluated by observing its supplement to angle 1 marked on the form. Since the sum of supplementary angles is \(180^{\circ}\) we can say, $$\alpha = 180 - 1 = 179^{\circ}$$

Determine the value of x

Since \(\alpha = x\), we can substitute the value of \(\alpha\) to find x: $$ x = 179^{\circ}$$ So, the value of x isn't within any of the given options. There might be an issue with the problem statement or the diagram.

Key concepts

Alternate Interior Angles

Alternate interior angles are angles that form on opposite sides of a transversal cutting through two parallel lines. They lie inside the parallel lines, which means they are positioned between the lines. These angles are termed 'alternate' because each pair lies on opposite sides of the transversal. A key feature of alternate interior angles is that they are always congruent.

For instance, suppose we have two parallel lines, \(\overline{DE}\) and \(\overline{AC}\), cut by a transversal, \(\overline{AB}\). If \(\angle\) 3 below the transversal on line \(\overline{DE}\) has the same measure as \(\angle\) 2 above the transversal on line \(\overline{AC}\), then they are alternate interior angles. This means \(\angle\) 3 and \(\angle\) 2 are of equal measure.

A fundamental property of alternate interior angles is that when parallel lines are involved, these angles are congruent. This means if you know the measure of one alternate interior angle, the other is exactly the same. This principle helps establish equations in geometry problems, like finding missing values of angles.

Congruent Angles

Congruent angles are angles that have exactly the same measure. In geometrical problems involving parallel lines and transversals, congruent angles often result from angle relationships such as alternate interior angles or corresponding angles.

Consider when parallel lines are cut by a transversal, several pairs of angles are formed. Among these, certain pairs are congruent. For example, alternate interior angles or corresponding angles will have the same measure due to the properties of parallel lines.

• Alternate Interior Angles: Found on opposite sides of the transversal and equal, as explained earlier.
• Corresponding Angles: Occupy the same relative position at intersection points of a transversal with each parallel line, and are equal in measure.

Knowing that specific angles are congruent can simplify the process of solving geometric problems. For instance, if two angles are found to be alternate interior and hence congruent, we can directly set their measures equal in any geometric equations utilised to find unknown angles.

Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees. This concept is crucial when analyzing angles formed by parallel lines and transversals. When two lines intersect, they form pairs of supplementary angles known as linear pairs. These pairs lie on the same side of a transversal or are adjacent to one another.

A real-world application of supplementary angles is when determining an unknown angle measure given its supplement. In the context of two intersecting lines making up supplementary angles, if one angle measure is known, simply subtract that measure from 180 to find the other. This property is often used in conjunction with alternate interior angles to solve problems involving parallel lines and transversals.

Take, for instance, determining angle \(\alpha\) in a problem where \(\angle\) 1 and \(\alpha\) are supplementary: these two angles together should equal 180 degrees, implying \(\alpha = 180 - 1\). Understanding supplementary angles facilitates solving geometric puzzles, especially when paired with transversal angle properties.