What is the solution to the inequality below?
1 answer:
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The values of the angles are; x = 12°, m∠PQS = 71°, m∠PQT = 142° and m∠TQR = 41°
<h3>What are the measure of the each angle?</h3>The required angles; x, m∠PQS, m∠PQT, and m∠TQR
The given parameters are bisects ∠PQT
m∠SQT = (8·x - 25)°
m∠PQT = (9·x + 34)°
m∠SQR = 112°
We have;
m∠PQT = m∠SQT + m∠PQS (Angle addition postulate)
m∠SQT ≅ m∠PQS (Angles formed by angle bisector are congruent)
m∠SQT = m∠PQS by Definition of congruency
m∠PQT = 2 × m∠SQT
Therefore;
(9·x + 34)° = 2 × (8·x - 25)° = (16·x - 50)°
Collecting like terms gives:
(34 + 50)° = 16·x - 9·x = 7·x
7·x = 84°
x = 84°/7 = 12°
x = 12°
m∠SQT = (8·x - 25)°
Therefore;
m∠SQT = (8 × 12 - 25)° = 71°
m∠SQT = 71°
m∠PQT = 2 × m∠SQT
∴ m∠PQT = 2 × 71° = 142°
m∠PQT = 142°
m∠PQS = m∠SQT (Angles formed by the same bisector )
∴ m∠PQS = m∠SQT = 71°
m∠PQS = 71°
m∠SQR = m∠SQT + m∠TQR (Angle addition postulate)
m∠SQT = 71°
∴ m∠SQR = 112° = 71° + m∠TQR
m∠TQR = 112° - 71° = 41°
m∠TQR = 41°
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Short Answers:
Answer for part A: Definition of perpendicular
Answer for part B: Right Angle Congruence Theorem
Answer for part C: Reflexive Property of Congruence
Answer for part D: Definition of Midpoint
Answer for part E:
Answer for part F: CPCTC
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Explanations:
Part A:
We are given that
which means, in english, "line segment RX is perpendicular to line segment ST"
By the very definition of perpendicular, this means that the two line segments form a right angle. This is visually shown as the red square angle marker for angle RXT. Angle RXS is also a right angle as well.
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Part B:
The Right Angle Congruence Theorem (aka Right Angle Theorem) is the idea that if we have two right angles, then we know that they are both 90 degrees so they must be congruent to one another.
---------------------
Part C:
Any line segment is congruent to itself. This is because any line segment will have the same length as itself. It seems silly to even mention something so trivial but it helps establish what we need for the proof.
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Part D:
We are given "X is the midpoint of segment ST" so by definition, X is in the very exact middle of ST. Midpoints cut segments exactly in half. SX is one half while TX is the other half. The two halves are congruent which is why SX = TX
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Part E:
Writing means "triangle SXR is congruent to triangle TXR". These two triangles are the smaller triangles that form when you draw in segment RX
Side Note: SAS stands for "side angle side". The angle must be between the two sides. The pairing RX and RX forms one of the 'S' letters (see part C), while the pairing SX and TX forms the other 'S' (see part D). The angles between the sides are RXS and RXT (see part B).
---------------------
Part F:
CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent"
It means that if we have two congruent triangles, then the corresponding parts are congruent. Back in part E, we proved the triangles congruent. For this part, we look at the pieces RS and RT (which correspond to one another; they are the hypotenuse of each triangle). They are proven congruent by CPCTC
If CPCTC is an odd concept to think about, then try thinking about something like this: you have two houses which are completely identical in every way. We can say that those two houses are congruent. If the houses are identical, then surely every piece that makes up the house is identical to its corresponding piece to the other house. For example, the front door to each house is both the same size, shape, color, made of the same material, designed in the same pattern, etc. So the two doors are congruent as well.
Answer for part A: Definition of perpendicular
Answer for part B: Right Angle Congruence Theorem
Answer for part C: Reflexive Property of Congruence
Answer for part D: Definition of Midpoint
Answer for part E:
Answer for part F: CPCTC
-------------------------------------------------------------
Explanations:
Part A:
We are given that
By the very definition of perpendicular, this means that the two line segments form a right angle. This is visually shown as the red square angle marker for angle RXT. Angle RXS is also a right angle as well.
---------------------
Part B:
The Right Angle Congruence Theorem (aka Right Angle Theorem) is the idea that if we have two right angles, then we know that they are both 90 degrees so they must be congruent to one another.
---------------------
Part C:
Any line segment is congruent to itself. This is because any line segment will have the same length as itself. It seems silly to even mention something so trivial but it helps establish what we need for the proof.
---------------------
Part D:
We are given "X is the midpoint of segment ST" so by definition, X is in the very exact middle of ST. Midpoints cut segments exactly in half. SX is one half while TX is the other half. The two halves are congruent which is why SX = TX
---------------------
Part E:
Writing
Side Note: SAS stands for "side angle side". The angle must be between the two sides. The pairing RX and RX forms one of the 'S' letters (see part C), while the pairing SX and TX forms the other 'S' (see part D). The angles between the sides are RXS and RXT (see part B).
---------------------
Part F:
CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent"
It means that if we have two congruent triangles, then the corresponding parts are congruent. Back in part E, we proved the triangles congruent. For this part, we look at the pieces RS and RT (which correspond to one another; they are the hypotenuse of each triangle). They are proven congruent by CPCTC
If CPCTC is an odd concept to think about, then try thinking about something like this: you have two houses which are completely identical in every way. We can say that those two houses are congruent. If the houses are identical, then surely every piece that makes up the house is identical to its corresponding piece to the other house. For example, the front door to each house is both the same size, shape, color, made of the same material, designed in the same pattern, etc. So the two doors are congruent as well.