There are only a couple of Lines and Angles questions tested on the ACT Math section. The questions are usually easy and should not pose a challenge to you.
∠2 = ∠4
∠5 = ∠7
∠6 = ∠8
∠1 = ∠5
∠2 = ∠6
∠3 = ∠7
∠4 = ∠8
∠1 = ∠7
∠2 = ∠8
∠4 = ∠6
∠3 = ∠5
So, putting all information together:
∠1, 3, 5 and 7 are congruent;
∠2, 4, 6 and 8 are congruent.
The opposite of all of this is also true: if you don't know if two lines are parallel - look at the alternate interior or exterior angles. If they are congruent, the lines are parallel.
Refer to the figure and two statements. Evaluate ∠1 + ∠2.
A. 82°
B. 98°
C. 180°
D. 262°
Solution: Let’s consider the first statement by itself. ∠2, ∠3, ∠4, and ∠5 together form a straight angle, so their degree measures a total 180°.
∠2 + ∠3 + ∠4 + ∠5 = 180°
∠2 + 131° = 180°
∠2 = 49°
Without further information, no other angle measures, including that of ∠1, can be found. So, let’s consider the second statement. ∠1, ∠6, ∠7, and ∠8 together form a straight angle, so their degree measures a total 180°.
∠1 + ∠6 + ∠7 + ∠8 = 180°
∠1 + 131° = 180°
∠1 = 49°
We’ve found both ∠1 and ∠2, so all we have to do is add them together to solve the problem:
∠1+ ∠2 = 49° + 49° = 98°
The correct answer is B.
Given the above figure and two statements, which of the following is true?
A. Line m is perpendicular to line t
B. Line m is not perpendicular to line t
C. ∠4 = ∠7
D. ∠1 = ∠6
Solution: Statement A alone establishes by definition that line l is not perpendicular to line t, but it does not establish any relationship between line m and line t.
By Statement B alone, we know line l is parallel to line m because alternating interior angles are congruent., line l is parallel to line m. ∠2 and ∠6 are corresponding angles formed by a transversal across parallel lines, so ∠6 = ∠2 = 89°. ∠6 is not a right angle, so line m is not perpendicular to line t and B is the correct answer.
Consider the line diagram shown at right. Which of the following pieces of information is enough to prove that AB is parallel to CD?
A. ∠1 = ∠3
B. ∠4 = 120° and ∠8 = 60°
C. ∠5 = 120° and ∠6 = 60v
D. ∠2 = ∠6 = 100°
Solution: By learning that ∠4 = 120° and ∠8 = 60°, we have enough information to determine that ∠1 = 60° (because it forms a straight angle with ∠4 ) and that ∠7 = 120° (since it forms a straight angle with ∠8 ). At this point, we can see that the pairs of corresponding angles, angles 1 and 8 and angles 4 and 7, match in value. This means that AB is parallel to CD. The correct answer choice is B.
∠ABD = 60°. What is ∠DBE?
A. 30°
B. 45°
C. 60°
D. 90°
Solution: The sum of ∠ABD, ∠DBE, and ∠EBC equally 180° because they are all on the straight-line AC.
∠ABD + ∠DBE + ∠EBC = 180°
We know that ∠ABD = 60° and that ∠EBC = 90°, so we can substitute in those values.
60° + ∠DBE + 90° = 180°
∠DBE = 30°
The correct answer is A.
The rotation from the initial side to the terminal side is a measure of an angle. The counterclockwise rotation is considered positive. An angle in the clockwise direction is taken as negative. CO-TERMINAL angles are the two angles that have the same terminal side.
RADIANS and DEGREES are units to measure angles. The conversion between these two units can be done using the following equation.
\(\frac{1}{360} degree=\frac{1}{2\pi}radian\) OR
\(\frac{1}{180} degree=\frac{1}{\pi}radian\)
To convert from degrees to radians and vice versa, we need the following:
Deg. = Rad. x \(\frac{180}{\pi}\)
Rad. = Deg. x \(\frac{\pi}{180}\)
For example, if we need to convert 50 degrees to radians, we need to multiply the degree measure by \(\frac{\pi}{180}\) ,
i.e. 50°= 50 x \(\frac{\pi}{180}rad = \frac{5\pi}{18}rad\)
Similarly, if we need to convert radians into degrees, we need to multiply the radian measure by \(\frac{\pi}{180}\)
Some important conversions are:
Example: In the xy-plane, O is the center of the circle, and the measure of radians. What is the value of a?
By the distance formula, the length of radius OA is \(\sqrt{\sqrt{3^{2}}+1^{2}}=2\)
Thus, SIN (∠AOB) = \(\frac{1}{2}\)
Therefore ∠AOB is 30°, which is equal to \(30\frac{(\pi)}{180}=\frac{\pi}{6}radians\)
Hence, the value of a is 6.
Circles and Conics questions on the ACT are straightforward and formula driven. The following chapters will list all the formulas you need to know to answer the circle questions on the ACT Math section.
There are only a couple of Lines and Angles questions tested on the ACT Math section. The questions are usually easy and should not pose a challenge to you.
The ACT is absolutely obsessed with triangle-based problems, and as a result, the ACT Math section includes many questions on the ACT Math section. The following set of formulas will help eradicate your trouble once and for all.
The word TRIGONOMETRY is Greek; tri means three, gon means angle and meter means to measure. Combining all these three words, we get, “three angle measure”. The logic behind this is that trigonometry involves the measurements inside a triangle, which has three angles.
The polygons tested on the ACT are usually the following quadrilaterals: trapezoids, parallelograms, and special parallelograms, such as rhombuses, rectangles, and squares.
SOLID GEOMETRY questions on the ACT Math section are some of the hardest questions you will see on the ACT Math section.
COORDINATE GEOMETRY is the study of geometry on a plane using a coordinate system. This plane is called a COORDINATE PLANE, which has scales of measurement along the x and y-axes.
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