Trapezoid and Kite Worksheet - Free Printable

Let’s solve each problem step by step.

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Problem 1: Figure with points F, G, H, I — looks like a trapezoid (one pair of parallel sides). Given angle at I is 110°. Marks on FG and HI suggest they are the non-parallel legs? Wait — actually, looking at arrows: FI and GH have one arrow → so those are parallel. So it's a trapezoid with bases FI and GH.

In a trapezoid with one pair of parallel sides (FI ∥ GH), consecutive angles between the parallel sides are supplementary (add to 180°).

So:

- Angle I = 110° → then angle H (adjacent along leg IH) should be 180° - 110° = 70°
- Similarly, since FI ∥ GH, angle F + angle G = 180°? But we don’t know either yet.

Wait — but look at tick marks: FG and HI both have one tick mark → meaning FG ≅ HI. So this is an isosceles trapezoid!

In an isosceles trapezoid:
- Base angles are equal.
- Angles adjacent to each base are equal.
- Consecutive angles between the bases are supplementary.

Given: ∠I = 110°

Since FI ∥ GH, then ∠I + ∠H = 180° → ∠H = 70°

In isosceles trapezoid, base angles are equal → so ∠F = ∠I = 110°, and ∠G = ∠H = 70°

Check: sum of interior angles = 360° → 110+70+110+70 = 360 ✔️

✔ So:

name: Isosceles Trapezoid

m∠F = 110°

m∠G = 70°

m∠H = 70°

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Problem 2: Quadrilateral J K L M — all sides marked with two ticks → all sides equal → rhombus or kite? Also, diagonals? Not drawn. But angles given: ∠M = 98°, ∠K = 120°. And sides MJ=JK=KL=LM? All four sides equal → must be a rhombus.

But wait — in a rhombus, opposite angles are equal, and consecutive angles are supplementary.

Here, ∠M = 98°, ∠K = 120° — these are NOT equal → contradiction if it were a rhombus.

Wait — look again: sides MJ and JK have two ticks, KL and LM also have two ticks — but maybe MJ = JK and KL = LM, but not necessarily all four equal? Actually, from diagram: MJ and JK share vertex J, both have double tick; KL and LM share vertex L, both have double tick — so likely MJ = JK and KL = LM, but MJ may not equal KL. That makes it a kite.

Yes! A kite has two pairs of adjacent congruent sides.

Properties of kite:
- One diagonal is axis of symmetry.
- One pair of opposite angles are equal (the ones between unequal sides).
- The other two angles are not necessarily equal.
- Sum of interior angles = 360°.

Given: ∠M = 98°, ∠K = 120°

In kite J-K-L-M, assuming vertices in order, then angles at J and L are the “vertex” angles? Or which are equal?

Actually, in a kite, the angles between the unequal sides are equal. Let’s assume the kite is symmetric across diagonal JL or KM.

Standard labeling: usually, the two equal angles are the ones that are NOT between the pairs of equal sides.

But here, since MJ = JK and KL = LM, then the equal angles should be at J and L? Or at M and K?

Wait — better approach: use angle sum.

Sum of angles = 360°

We have ∠M = 98°, ∠K = 120° → total so far = 218°

Remaining for ∠J + ∠L = 360 - 218 = 142°

In a kite, typically one pair of opposite angles are equal — specifically, the angles where the equal sides meet. Since MJ = JK, angle at J is between two equal sides → similarly, KL = LM, so angle at L is between two equal sides → so ∠J = ∠L.

Therefore, ∠J = ∠L = 142° / 2 = 71°

But the question asks for m∠K and m∠I — wait, there is no point I! Typo? Looking back: problem says "find measure of angles" and lists m∠K and m∠I — but figure has J, K, L, M. Probably typo — should be m∠J and m∠L? Or maybe m∠K and m∠M? But those are given.

Wait — re-read: “Name the figure and find the measure of angles.” Then it says:

name _______________

m∠K = _______ m∠I = _______

But there is no point I in the figure. Must be a typo. Likely meant m∠J and m∠L? Or perhaps m∠M and m∠L? But ∠M is given as 98°.

Looking at the figure description: points J, K, L, M. Probably “m∠I” is a mistake — should be “m∠J” or “m∠L”.

But in many textbooks, sometimes they label differently. Alternatively, maybe “I” is a typo for “J”? Because J is next to I alphabetically? Unlikely.

Another possibility: in some diagrams, they might have labeled differently, but based on standard, let’s assume it’s asking for the unknown angles: ∠J and ∠L.

And since it’s a kite with MJ=JK and KL=LM, then ∠J = ∠L = 71° as above.

But the problem writes “m∠K = ___ m∠I = ___” — perhaps “I” is meant to be “J”? Or “L”?

Wait — looking at the original image description: user said “Figure with points J, K, L, M” — and in the text, it says “m∠K = ___ m∠I = ___” — definitely a typo. Most likely, it should be m∠J and m∠L, or perhaps m∠M and m∠L — but ∠M is given.

Perhaps “I” is a misprint for “J”. I’ll go with that.

So:

name: Kite

m∠K = 120° (given)

m∠J = 71° (calculated)

But the blank says m∠I — if we assume it’s a typo and should be m∠J, then 71°.

Alternatively, maybe the figure has point I? No, user described only J,K,L,M.

To resolve: in the answer, I’ll put m∠K = 120°, and for m∠I, since it doesn't exist, but likely meant m∠J = 71°.

But let’s check: perhaps in some systems, they label the fourth point as I? Unlikely.

Another thought: maybe “I” is a typo for “L”? Then m∠L = 71°.

Either way, the unknown angles are 71° each.

I think safest is to state:

name: Kite

m∠K = 120° (given)

m∠J = 71° (assuming "I" is typo for "J")

But to match the blank, perhaps write m∠I as 71°, noting it's likely a typo.

For now, I'll proceed with:

name: Kite

m∠K = 120°

m∠I = 71° [assuming typo for ∠J or ∠L]

But let's confirm calculation:

Angles: M=98, K=120, J=?, L=?

Sum = 360

If J=L, then 98+120+2x=360 → 218+2x=360 → 2x=142 → x=71. Yes.

And in kite, the angles between the equal sides are equal — here, at J (between MJ and JK, which are equal) and at L (between KL and LM, which are equal), so yes, ∠J = ∠L = 71°.

So final for problem 2:

name: Kite

m∠K = 120°

m∠I = 71° [with note that "I" is likely a typo for "J" or "L"]

But since the problem says m∠I, and there's no I, I'll still put 71°, assuming it's a labeling error.

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Problem 3: Rectangle PQRS. Diagonals intersect at T. Given ∠SPQ = 82°? Wait, in rectangle, all angles are 90°. But it says m∠SPQ = 82°? That can't be right for a rectangle.

Look: "Given rectangle PQRS", and diagram shows angle at P is 82°? But in a rectangle, every angle is 90°. Contradiction.

Unless... perhaps it's not the corner angle? Diagram shows diagonal PR and QS intersecting at T, and angle at P between SP and PT is 82°? Ah! Probably ∠SPT = 82°, not ∠SPQ.

Re-reading: "m∠SPQ = ___" — but in the diagram, it might be labeled as angle between side and diagonal.

The text says: "m∠SPQ = ___" — but in rectangle, ∠SPQ should be 90°. Unless it's a typo.

Looking at the diagram description: "angle at P is 82°" — likely it's ∠SPT = 82°, where T is intersection of diagonals.

Because in rectangle, diagonals bisect each other and are equal, so triangles formed are isosceles.

Assume that the 82° is ∠SPT, i.e., angle between side SP and diagonal PT.

In rectangle PQRS, let’s say P is bottom-left, Q bottom-right, R top-right, S top-left. Diagonals PR and QS intersect at T.

Then triangle PST: PS is side, PT is half-diagonal.

Given ∠SPT = 82°.

Since PQRS is rectangle, ∠SPQ = 90° (corner angle).

Diagonal PR divides ∠SPQ into ∠SPT and ∠TPQ.

So if ∠SPT = 82°, then ∠TPQ = 90° - 82° = 8°.

Now, in rectangle, diagonals are equal and bisect each other, so PT = QT = RT = ST.

Thus, triangle PST is isosceles with PT = ST.

Similarly, triangle PQT is isosceles with PT = QT.

First, find angles in triangle PST.

We have ∠SPT = 82°.

Since PT = ST, then ∠PST = ∠SPT = 82°? No, in triangle PST, sides PT and ST are equal, so base angles are equal. Vertices P, S, T.

Sides: PT and ST are equal (since diagonals bisect each other and are equal, so half-diagonals are equal).

So in triangle PST, PT = ST, so angles opposite them are equal: angle at S and angle at P.

Angle at P is ∠SPT = 82°, so angle at S, which is ∠PST, should also be 82°.

Then angle at T, ∠PTS = 180° - 82° - 82° = 16°.

But is that correct? Let me sketch mentally.

Point P, S, T. Side PS is vertical (say), PT is diagonal to center.

If ∠SPT = 82°, and PT = ST, then yes, triangle PST has PT = ST, so ∠PST = ∠SPT = 82°, so ∠PTS = 16°.

Now, the questions:

m∠PST = ? → that's angle at S in triangle PST, which is 82°.

m∠PTS = ? → angle at T in triangle PST, which is 16°.

m∠QPR = ? → angle at P in triangle QPR? Or angle between QP and PR.

∠QPR is part of corner angle at P.

At vertex P, total angle is 90° between SP and QP.

Diagonal PR splits it into ∠SPR and ∠QPR.

We have ∠SPT = 82°, and T is on PR, so ∠SPR = ∠SPT = 82°, thus ∠QPR = 90° - 82° = 8°.

Similarly, m∠PTQ = ? → angle at T in triangle PTQ.

Triangle PTQ: points P, T, Q.

PT = QT (half-diagonals), so isosceles.

Angle at P in this triangle is ∠QPT = ∠QPR = 8°.

Since PT = QT, then angles at P and Q are equal? In triangle PTQ, sides PT and QT are equal, so base angles at P and Q are equal.

Vertex T, so angles at P and Q are the base angles.

So ∠QPT = ∠PQT = 8°.

Then angle at T, ∠PTQ = 180° - 8° - 8° = 164°.

m∠SRT = ? → angle at R in triangle SRT.

First, by symmetry, similar to triangle PST.

At vertex R, corner angle is 90°.

Diagonal PR and SR.

Triangle SRT: S, R, T.

ST = RT (half-diagonals), so isosceles.

Angle at R: ∠SRT.

Note that ∠SRP is part of corner angle at R.

By symmetry, since rectangle is symmetric, and we had at P, ∠SPR = 82°, so at R, ∠SRP should be the same? Let's see.

Actually, in rectangle, diagonal PR creates two congruent triangles: ΔPQR and ΔPSR? No, ΔPQS and ΔQRS? Better: triangles formed by diagonal.

Triangle SPR and triangle QRP are congruent? Perhaps.

From earlier, at P, ∠SPR = 82°, so in triangle SPR, which is right-angled at S? No.

Triangle SPR: points S, P, R. Angle at S is 90°, since rectangle.

In triangle SPR, angle at S is 90°, angle at P is ∠SPR = 82°, so angle at R, ∠SRP = 180° - 90° - 82° = 8°.

Is that right? Triangle SPR: vertices S, P, R. Side SP and SR are adjacent sides, so angle at S is 90°. Diagonal PR.

So yes, in triangle SPR, angles: at S: 90°, at P: 82°, so at R: 8°.

Thus, ∠SRP = 8°.

Now, T is midpoint of PR, so in triangle SRT, we have points S, R, T.

ST and RT are half-diagonals, so ST = RT.

Angle at R in triangle SRT is ∠SRT, which is part of ∠SRP.

Since T is on PR, and ∠SRP = 8°, and T is between P and R, so ∠SRT = ∠SRP = 8°? Only if T is close to R, but actually, since it's the same ray, yes, ∠SRT is the same as ∠SRP, because it's the angle between SR and RT, and RT is along RP.

Ray RT is the same as ray RP, since T is on PR.

So ∠SRT = ∠SRP = 8°.

Similarly, in triangle SRT, ST = RT, so it's isosceles with ST = RT, so base angles at S and R are equal.

Angle at R is 8°, so angle at S, ∠RST = 8°.

Then angle at T, ∠STR = 180° - 8° - 8° = 164°.

But the question is m∠SRT, which is angle at R, so 8°.

Now, m∠QRE? What is E? Probably typo. Should be m∠QRT or something.

Look: "m∠QRE = ___" — no point E. Likely typo. Perhaps m∠QRT or m∠QTR.

In context, probably m∠QRT or m∠TQR.

Earlier we have triangle PTQ, and we found ∠PTQ = 164°, etc.

Perhaps "E" is a typo for "T", so m∠QRT.

Point Q, R, T.

Triangle QRT: Q, R, T.

QT = RT (half-diagonals), so isosceles.

Angle at R: ∠QRT.

At vertex R, total angle is 90° between QR and SR.

We have ∠SRP = 8°, and since ∠SRQ = 90°, then ∠QRP = 90° - 8° = 82°.

Because diagonal PR splits the 90° angle at R into ∠SRP and ∠QRP.

We have ∠SRP = 8°, so ∠QRP = 82°.

Now, T is on PR, so in triangle QRT, angle at R is ∠QRT = ∠QRP = 82°.

Since QT = RT, triangle QRT is isosceles with QT = RT, so base angles at Q and R are equal? Sides QT and RT are equal, so angles opposite them: angle at R and angle at Q.

Opposite QT is angle at R, opposite RT is angle at Q.

So ∠QRT = ∠RQT.

We have ∠QRT = 82°, so ∠RQT = 82°.

Then angle at T, ∠QTR = 180° - 82° - 82° = 16°.

But the question is m∠QRE — if E is T, then m∠QRT = 82°.

Perhaps it's m∠QTR, but it says QRE.

Another possibility: "E" might be a typo for "S", but unlikely.

Or perhaps in some labels, but I think most likely it's m∠QRT = 82°.

To confirm, let's list what we have:

From above:

m∠PST = 82° (in triangle PST, angle at S)

m∠PTS = 16° (angle at T in triangle PST)

m∠QPR = 8° (angle at P between QP and PR)

m∠PTQ = 164° (angle at T in triangle PTQ)

m∠SRT = 8° (angle at R in triangle SRT)

m∠QRE — assume m∠QRT = 82°

Now, types of triangles:

ΔPST: we have angles 82°, 82°, 16° — so isosceles triangle (two angles equal)

ΔPSR: points P, S, R — this is a right triangle, since angle at S is 90°. Angles: at S 90°, at P 82°, at R 8° — so scalene right triangle, but specifically, it's a right triangle.

The question is: "What type of triangle is ΔPST? ________ ΔPSR? ________"

ΔPST: as above, isosceles (since PT = ST, and angles at P and S equal)

ΔPSR: this is triangle P-S-R, which includes the diagonal. As above, it's a right triangle with angles 90°, 82°, 8° — so it's a right triangle, and since all angles different, scalene, but typically we say "right triangle".

In geometry problems, often they want "right", "acute", "obtuse", or "isosceles", etc.

ΔPSR has a right angle at S, so it's a right triangle.

ΔPST has two equal angles, so isosceles.

Now, for the blanks:

m∠PST = 82°

m∠PTS = 16°

m∠QPR = 8°

m∠PTQ = 164°

m∠SRT = 8°

m∠QRE — I'll assume 82° for m∠QRT

Types:

ΔPST: isosceles

ΔPSR: right triangle

But let's double-check m∠QRE. Perhaps it's m∠QTR or something else.

Another thought: in some diagrams, E might be a point, but here no. Perhaps "E" is a typo for "T", and it's m∠QRT.

Maybe it's m∠QTS or something.

To be safe, I'll go with 82°.

So summary for problem 3:

m∠PST = 82°

m∠PTS = 16°

m∠QPR = 8°

m∠PTQ = 164°

m∠SRT = 8°

m∠QRE = 82° [assumed]

ΔPST: isosceles

ΔPSR: right triangle

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**Problem 4: Square EFGH. Diagonals intersect at J. Given angle at G is 15°? Diagram shows angle between EG and HG is 15°? But in square, diagonals bisect the corner angles, so should be 45°.

Read: "Given square EFGH, find the measure of the angles." Diagram has angle at G between diagonal EG and side HG labeled 15°? That can't be for a square.

Unless it's not the corner angle. Probably, the 15° is ∠EGH or something.

Text says: "m∠EJF = ___" etc., and diagram has "15" near G, likely ∠HGE = 15° or something.

Assume that in square EFGH, diagonals intersect at J. The 15° is probably ∠FHG or ∠EGH.

Standard: in square, diagonals bisect the 90° angles, so each half is 45°. But here it's given as 15°, so perhaps it's not the bisected angle.

Perhaps the 15° is the angle between the diagonal and the side, but in square it should be 45°.

Unless the square is labeled differently.

Another possibility: the 15° is ∠JGH or something.

Look at the text: "m∠EJF = ___" — angle at J in triangle EJF.

In square, diagonals are perpendicular and bisect each other, so at intersection J, angles are 90°.

Also, diagonals bisect the vertex angles, so at each corner, the diagonal makes 45° with the sides.

But here, diagram shows "15" at G, so likely ∠HGF is not 90°, but that can't be for a square.

Perhaps it's a typo, and it's 45°, but it's written as 15.

Or perhaps the 15° is for a different angle.

Another idea: perhaps "15" is the measure of ∠FGJ or something.

Let's read the problem: "Given square EFGH, find the measure of the angles." and "m∠EJF = ___" etc.

Diagram has points E,F,G,H, diagonals EG and FH intersect at J, and at vertex G, between side HG and diagonal EG, it's labeled 15°.

But in a square, that should be 45°. So unless it's not a square, but the problem says "square".

Perhaps it's a trick, or perhaps the 15° is for a different purpose.

Maybe the 15° is ∠JGH, and we need to find others.

Assume that in square EFGH, at vertex G, the diagonal EG makes an angle of 15° with side HG. But that would mean the corner angle is not 90°, contradiction.

Unless the square is rotated, but still, the angle between diagonal and side is always 45° in a square.

So probably, the 15° is a red herring or typo. Perhaps it's 45°, but written as 15 by mistake.

Maybe "15" is the length or something, but it's placed at the angle.

Another possibility: the 15° is ∠EHG or something, but at H.

Let's look at the angles asked: m∠EJF, m∠HEG, etc.

In square, diagonals intersect at 90°, so m∠EJF = 90°, since it's the angle at intersection.

m∠HEG: angle at E in triangle HEG. Points H,E,G. Diagonal EG, so in triangle HEG, which is half the square.

In square, triangle HEG is isosceles right triangle, with right angle at H? No.

Vertices H,E,G: if E and G are opposite corners, then HG and HE are sides, EG diagonal.

So triangle HEG has sides HE, HG (sides of square), and EG (diagonal).

Angle at H is 90°, since corner of square.

Angles at E and G are 45° each, because diagonals bisect the angles.

So m∠HEG = 45°.

Similarly, m∠EHG = 90°, but not asked.

The problem gives "15" at G, so perhaps for this problem, it's not a standard square, or perhaps the 15° is for a different angle.

Perhaps the 15° is ∠FGJ or ∠JGH.

Assume that the 15° is ∠HGE, i.e., angle between HG and GE.

But in square, it should be 45°, so if it's given as 15°, then it's not a square, but the problem says "square".

This is confusing.

Perhaps "15" is the measure of ∠JGH, and we need to use that.

Let's try to interpret.

Suppose in square EFGH, diagonals intersect at J. At vertex G, the diagonal EG is drawn, and the angle between side HG and diagonal EG is given as 15°. But that would imply that the corner angle at G is not 90°, which contradicts square.

Unless the 15° is the angle between the diagonal and the extension or something.

Another idea: perhaps the 15° is ∠EGH, but in the triangle.

Let's calculate based on properties.

In square, regardless, diagonals are perpendicular, so m∠EJF = 90°.

m∠HEG: as above, in triangle HEG, angle at E is 45°.

m∠EHG: angle at H in triangle EHG, which is 90°.

m∠FEG: angle at E in triangle FEG. Points F,E,G. FE and EG.

At vertex E, the diagonal EG bisects the 90° angle, so ∠FEG = 45°.

Similarly, m∠EFG = 90°, m∠FGE = 45°.

m∠FGE is angle at G in triangle FEG, which is 45°.

But the diagram has "15" at G, so perhaps for this problem, the 15° is given to indicate that it's not 45°, but that doesn't make sense.

Perhaps "15" is the measure of ∠JGH, and J is intersection, so in triangle JGH.

In square, triangle JGH is isosceles right triangle, with angles 45°, 45°, 90° at J.

At G, angle is 45°.

If it's given as 15°, then perhaps it's a different shape, but the problem says "square".

I think there might be a typo in the problem or diagram. Perhaps the 15° is for a different problem, or it's 45°.

Maybe "15" is the length of a side, but it's placed at the angle.

Another possibility: the 15° is ∠FHG or something else.

Let's look at the angles asked: m∠EJF, m∠HEG, m∠EHG, m∠FEG, m∠EFG, m∠FGE.

In a square, all these can be determined without additional information.

- m∠EJF = 90° (diagonals perpendicular)

- m∠HEG = 45° (diagonal bisects corner angle)

- m∠EHG = 90° (corner angle)

- m∠FEG = 45° (same as ∠HEG, since symmetric)

- m∠EFG = 90° (corner angle)

- m∠FGE = 45° (bisected angle)

Then types of triangles:

ΔEHG: points E,H,G — this is a right triangle with right angle at H, and legs equal, so isosceles right triangle.

ΔEJH: points E,J,H — J is intersection of diagonals. In square, EJ = HJ, and angle at J is 90°, so isosceles right triangle.

So perhaps the 15° is irrelevant or a distractor, or for a different part.

Maybe the 15° is for ∠JGH, and we need to use it, but in square it should be 45°.

Perhaps the figure is not a square, but the problem says it is.

I think for the sake of solving, I'll assume it's a standard square, and the 15° is a mistake, or perhaps it's for another angle.

Maybe "15" is the measure of ∠EGH in degrees, but that would be 45°, so perhaps it's 45, and 15 is typo.

Or perhaps it's 45, and written as 15 by error.

Another idea: perhaps the 15° is the angle between the diagonal and the side, but in a different configuration.

Let's calculate if we take the 15° as given.

Suppose at vertex G, the angle between side HG and diagonal EG is 15°. Then, since the corner angle is 90°, the angle between side FG and diagonal EG would be 90° - 15° = 75°.

But in a square, the diagonal should make 45° with both sides, so this is impossible unless it's not a square.

Perhaps the "square" is mislabeled, and it's a rhombus or something.

But the problem explicitly says "square".

Perhaps the 15° is for ∠JGH, and J is not the intersection, but it is.

I think I have to go with the standard properties.

So for problem 4:

m∠EJF = 90° (diagonals intersect at 90°)

m∠HEG = 45° (diagonal bisects 90° angle)

m∠EHG = 90° (corner angle)

m∠FEG = 45° (same as ∠HEG)

m∠EFG = 90° (corner angle)

m∠FGE = 45° (bisected angle)

Then ΔEHG: isosceles right triangle (since EH = HG, angle at H 90°)

ΔEJH: isosceles right triangle (EJ = HJ, angle at J 90°)

So answers:

m∠EJF = 90°

m∠HEG = 45°

m∠EHG = 90°

m∠FEG = 45°

m∠EFG = 90°

m∠FGE = 45°

ΔEHG: isosceles right triangle

ΔEJH: isosceles right triangle

And ignore the 15° as likely a typo or for decoration.

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**Problem 5: Rhombus PQRS. Diagonals intersect at T. Given angle at P is 30°? Diagram shows ∠QPS = 30°, but in rhombus, opposite angles equal, consecutive supplementary.

Also, side QR = 6, but may not be needed for angles.

Given: ∠QPS = 30°.

In rhombus PQRS, assume P,Q,R,S in order.

So angle at P is ∠QPS = 30°.

Then, since consecutive angles are supplementary, angle at Q, ∠PQR = 180° - 30° = 150°.

Angle at R, ∠QRS = angle at P = 30° (opposite angles equal).

Angle at S, ∠RSP = angle at Q = 150°.

Now, diagonals intersect at T, and in rhombus, diagonals bisect the vertex angles and are perpendicular bisectors of each other.

So, diagonal PR bisects ∠QPS and ∠QRS.

So, ∠QPT = ∠SPT = 30° / 2 = 15°.

Similarly, diagonal QS bisects ∠PQR and ∠RSP, so ∠PQT = ∠RQT = 150° / 2 = 75°.

Now, the questions:

m∠PTS = ? — angle at T in triangle PTS.

First, in triangle PTS, or generally.

Diagonals intersect at T, and are perpendicular, so ∠PTQ = ∠QTR = ∠RTS = ∠STP = 90°.

Is that true? In rhombus, diagonals are perpendicular, so yes, at intersection T, all angles are 90°.

So m∠PTS = 90°? But let's see the notation.

∠PTS is angle at T in triangle PTS, which is the same as the angle between PT and TS, which is 90°, since diagonals perpendicular.

Similarly, m∠PST = ? — angle at S in triangle PST.

Points P,S,T.

In triangle PST, we have:

- At P: ∠SPT = 15° (as above, since diagonal bisects angle)

- At T: ∠PTS = 90° (diagonals perpendicular)

- So at S: ∠PST = 180° - 15° - 90° = 75°

Similarly, m∠RST = ? — angle at S in triangle RST.

Points R,S,T.

At S, the full angle is ∠RSP = 150°, and diagonal QS bisects it, so ∠RSQ = 75°, and since T is on QS, ∠RST = ∠RSQ = 75°.

In triangle RST, we can verify.

At S: ∠RST = 75°

At T: ∠RTS = 90° (diagonals perpendicular)

At R: ∠SRT = ?

First, at R, angle is ∠QRS = 30°, and diagonal PR bisects it, so ∠QRP = ∠SRP = 15°.

So in triangle RST, at R: ∠SRT = 15°

At T: 90°

At S: 75°

Sum: 15+90+75=180, good.

Now, m∠QRS = ? — this is the angle at R in the rhombus, which is 30°, as opposite to angle at P.

m∠QRP = ? — angle at R in triangle QRP, which is part of the bisected angle. Since PR bisects ∠QRS, and ∠QRS = 30°, so ∠QRP = 15°.

m∠RTS = ? — angle at T in triangle RTS, which is 90°, as diagonals perpendicular.

Now, types of triangles:

ΔPTS: points P,T,S. Angles: at P 15°, at T 90°, at S 75° — so scalene right triangle, but specifically, it's a right triangle.

ΔPQS: points P,Q,S. This is a triangle formed by two sides and a diagonal.

In rhombus, triangle PQS: sides PQ, QS, SP.

PQ = SP (sides of rhombus), so isosceles.

Angle at P is 30°, so in triangle PQS, angle at P is 30°, and since PQ = PS, then base angles at Q and S are equal.

Sum of angles: 30° + 2x = 180° → 2x = 150° → x = 75°.

So angles at Q and S are 75° each.

So it's an isosceles triangle.

Now, to summarize:

m∠PTS = 90°

m∠PST = 75°

m∠RST = 75°

m∠QRS = 30°

m∠QRP = 15°

m∠RTS = 90°

ΔPTS: right triangle (or scalene right, but probably "right triangle")

ΔPQS: isosceles triangle

Now, for the blanks.

Also, side QR = 6 is given, but not used for angles, so probably for another part, but here only angles are asked.

So final answers for problem 5:

m∠PTS = 90°

m∠PST = 75°

m∠RST = 75°

m∠QRS = 30°

m∠QRP = 15°

m∠RTS = 90°

ΔPTS: right triangle

ΔPQS: isosceles triangle

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Now, compiling all answers.

For problem 1:
name: Isosceles Trapezoid
m∠F = 110°
m∠G = 70°
m∠H = 70°

For problem 2:
name: Kite
m∠K = 120°
m∠I = 71° [assuming typo for ∠J or ∠L]

For problem 3:
m∠PST = 82°
m∠PTS = 16°
m∠QPR = 8°
m∠PTQ = 164°
m∠SRT = 8°
m∠QRE = 82° [assumed]
ΔPST: isosceles
ΔPSR: right triangle

For problem 4:
m∠EJF = 90°
m∠HEG = 45°
m∠EHG = 90°
m∠FEG = 45°
m∠EFG = 90°
m∠FGE = 45°
ΔEHG: isosceles right triangle
ΔEJH: isosceles right triangle

For problem 5:
m∠PTS = 90°
m∠PST = 75°
m∠RST = 75°
m∠QRS = 30°
m∠QRP = 15°
m∠RTS = 90°
ΔPTS: right triangle
ΔPQS: isosceles triangle

Now, for the final answer, since it's multiple parts, I'll list them clearly.

Final Answer:

1. name: Isosceles Trapezoid
m∠F = 110°
m∠G = 70°
m∠H = 70°

2. name: Kite
m∠K = 120°
m∠I = 71°

3. m∠PST = 82°
m∠PTS = 16°
m∠QPR = 8°
m∠PTQ = 164°
m∠SRT = 8°
m∠QRE = 82°
ΔPST: isosceles
ΔPSR: right triangle

4. m∠EJF = 90°
m∠HEG = 45°
m∠EHG = 90°
m∠FEG = 45°
m∠EFG = 90°
m∠FGE = 45°
ΔEHG: isosceles right triangle
ΔEJH: isosceles right triangle

5. m∠PTS = 90°
m∠PST = 75°
m∠RST = 75°
m∠QRS = 30°
m∠QRP = 15°
m∠RTS = 90°
ΔPTS: right triangle
ΔPQS: isosceles triangle